sun krishna ii

8/6/2019 Sun Krishna II

http://slidepdf.com/reader/full/sun-krishna-ii 1/17

18 1EEE TRANSACTIONS O N CIRCUITS AND SYSTEMS -11: ANALOG AN D DIGITAL SIGNAL PROCESSING. VOL. 39 . N O. I , JANUARY 1992

A Coding Theory Approach to Error Control

in Redundant Residue Number Systems-

Part 11: Multiple Error Detection

and CorrectionJenn-Dong Sun a n d H a r i K r i s h n a

Abstract-In this work, we study and extend the codingtheory approach to error control in redundant residue numbersystems (RRNS). We derive new computationally efficient algo-rithms for correcting multiple errors, single-burst error, anddetecting multiple errors. These algorithms reduce the computa-tional complexity of the previously kno wn algorithms by at leastan order of magnitude.

Index terms: Algorithms, multiple errors detection and cor-rection, burst residue errors, redundant residue number systems,coding theory, mixed radix conversion.

I. INTRODUCTION

MONG the earliest researchers, Szabo and Tanaka [ I ]A ave briefly sketched a method for single-error detec-

tion or single-error correction for use with a residue number

system (RNS). However, the error correction procedure given

in [ l ] is computationally inefficient. A lso, it appears to be

quite complicated fo r implementation. Watson and Hastings

[2] have constructed a redundant residue number system

(RRNS) to detect or correct single errors. However, their

method for error correction needs a correction table, which

may require large memory space, thereby making it impracti-

cal for the correction of more than single residue errors.

Therefore, multiple-residue-error correction has not been

investigated by them. Mandelbaum [3] showed how single-error correction can be accomplished in an RRNS code and

established that two redundant moduli with redundancy less

than the redundancy in [2] are necessary for single residue

digit error correction. Later, necessary and sufficient condi-

tions for minimal redundancy allowing the correction of the

whole class of single residue errors were derived by Barsi

and Maestrini [4], who a lso developed the concept of an RN S

product code (31, a concept that was earlier suggested by

Mandelbaum. Yau and Liu [6] designed two error-correction

algorithms, one for single residue-error correction and the

other for burst residue-error correction . Basically, the method

of Yau and Liu is Watson’s method with the error-correcting

Manuscript received December 4, 1990: revised July 3 1 , 1991. The work

of J . - D . Su n was supported by a fellowahip from Chung-Shan Insti tute ofScience and Technology, ROC. This paper was recommended by AssociateEditor E. J. Coyle.

The authors are with the Department of Electrical and Computer Engi-neering, Syracuse University, Syracuse, NY 13244.1240.

IEEE Log Number 9104716.

table replaced by appropriate computations. Consequently,

their implementation needs a memory space that is much

smaller than that required in [2]. Ramachandran [7] proposed

a method to correct single errors that establishes a trade-off

between computational complexity and extra redundant mod-

uli. A number of papers by Jenkins and his associates [8]-[ 111

applied mixed radix conversion (MRC) to digital filters and

residue number error checkers. Su and Lo [ 121 have used the

redundant digits of MRC as the entries to construct a lookup

table for single residue-error correction.In our previous paper [ 141, we developed a coding theory

approach to error control in RRNS. The concepts of Ham-

ming weight, minimum distance, weight distribution, and

error detection and correction capabilities in RRNS were

introduced. The necessary and sufficient conditions for the

desired error control capability were derived from the mini-

mum distance point of view. A special case generated the

maximum distance separable (MDS) RRNS. A computation-

ally efficient procedure was described for correcting single

residue error.

In this paper, we will extend the theory and present new

procedures for simultaneously correcting single error and

detecting multiple errors, and simultaneously correcting dou-

ble errors and detecting multiple errors. In addition, wepresent a procedure for correcting a single-burst error.

This paper consists of eight sections. In Section 11, we

cover the definitions and basic coding theory for RRNS. In

Section 111, we generalize the property of consistency check-

ing for RR NS. T hree computationally efficient procedures for

i ) simultaneously correcting single error and detecting m ulti-

ple errors, ii ) simultaneously correcting double errors and

detecting multiple errors, and i i i ) correcting single-burst er-

ror, are presented in Sections IV. V, and VI, respectively. In

Section VII, we extend the previous algorithms in [4] and [9]

for multiple-error correction and detection. Finally, we dis-

cuss and compare the various results in Section VIII. For

reasons explained in 1141, we focus exclusively on MDS-

RRNS in this paper.

11. D E F I N I T I O N SN D BASICC O D I N GHEORYOR RRNS

Let { r n , m z , . ., m k } be a set of k positive relatively

prime integers called nonredundant moduli and let their

1057-7130,9?$03.00 1992 IEEE



SUN AND KRISHNA: CODING THEORY APPROACH-PAKT 11 19

product be

Any integer X n the range [0, M ) can be uniquely repre-

sented by a k-tuple x = [x,, 2,. , x,],where

x,= X m o d m , ) .( 2 )

The integer x,, alled the ith residue digit, is the least

non-negative remainder obtained upon dividing X by m , .

The set of k-tuples and the interpretation function assigning

to each k-tuple an integer in the range [0, M ) , and vice

versa, define the residue number system (R NS) of moduli

m,, 2 , a , m k . From the Chinese remainder theorem

(CRT), for any given k-tuple [x,, 2,. ., x,]. where 0 5

x,< m,, there exists one and only one integer X such that

0 5 X < M and x,3 X mod m,). The numerical value of

X s computed using

k

I = I(3)= n x ,7 ' ,M, (mod M )

whereM

m iM, = - (4)

7 ; M i = 1 ( mo d m i ) . ( 5 )

For erro r control (both correction and d etection are included),

we are concerned with the redundant residue number system

(RRNS) obtained by appending n - k additional moduli,

m k f , , m k + Z , * m,, called the redundant moduli, to theRNS to form an RRNS of n positive pairwise relatively

prime moduli. The product n:= ,+ ,m, s denoted by M,. An

integer X in the range [0, M ) s represented as an n-tuple.

x = [x,, 2; . , x k , xk+ * ., x,], orresponding to nmoduli. The integers in the range [0, M ) n the RRNS are

called legitimate integers and the corresponding n-tuples arecalled legitimate, while the n-tuples associated with the

integers in the range [ M , MM,) are called illegitimate. The

integers in the range [ M , MM,) are defined as the illegiti-

mate numbers. In [14], we define the RRNS as an ( n , )semilinear code L?, as its codevectors satisfy the property of

linearity under certain appropriately predefined conditions.

All the legitimate numbers are valid, and the corresponding

residue vectors ar e said to constitute the k-dimensional code

space. Note that all the n-tuple residue representations form

an n-dimensional vector space. Every residue representation

in the code space is a codevector that can be divided into two

parts; the first k residue digits corresponding to the k

nonredundant moduli are called the information digits, and

the remaining n - k residue digits corresponding to the

n - k redundant moduli are called the parity digits.Some definitions and coding theorems in the RRNS [I41

will now be given in order to make this paper self-contained.

Definit ion 1: The Ha mm ing weight o fa vector x , wt( ) .

in an RNS is defined as the number of nonzero components

of x.

Definition 2: The H am min g dis tanc e between two code-

vectors xi and x i , d ( x , , x i) s the number of places in

which x i an d xi differ.

Definit ion 3: The mi n i mum d i s t ance d of the RRNS is

defined as

d = m i n { d ( x , , x i ) : x i , x j € L ? ,x i # x i } . (6)

For an RRNS code, the minimum distance d is the same asthe Hamming weight of the codevector in fl having the

smallest positive Hamming weight,

d = m i n { w t ( x ) : X E Q , x # O } . (7 )

Theorem I : The minimum distance of an RRNS code is d

if and only if the product of redundant moduli satisfies thefollowing relation:

For MDS-RRNS, d - 1 = n - k , and the moduli satisfy

M , = max{HYL, 'm, ,}, where 1I,I.Theorem 2: The error detecting capability, I , of an RRNS

Q is d - 1.

Theorem 3: The error correcting capability, t , of an

RRNS Q i s [ ( d - 1/2)], where [a ] denotes the largest

integer less than or equal to a .

Theorem 4: An RRNS is capable of correcting h or fewer

errors and simultaneously detecting /3 (/ 3 > h) or fewer

errors if , d 2 h + p + 1.

It will be seen that for an (n , k ) MDS-RRNS code, any

single or double-residue errors can be corrected, and /3 or

fewe r errors can be d etected if the RR NS satisfies the follow-

ing two conditions: 1) (Theorem 4 ) d - 1 = n - k = h + /3where h = 1 or 2 and /3 > A; nd 2 )

min {m, . ,mr ,} > m ax { 2 m , , m , , - m , , - mi l }

where k < r l , rzI an d 1 5 i , , i , 5 k. It is worthwhile

t o note here that the condition in 2) above is a sufficientcondition.

111.CONSISTENCYHECKINGOR RRNS

For an ( n , k ) MDS-RRNS code. assume that there are can d g errors (c + g 5 p ) in the information and parity

digits, respectively. Then the received residue vector can be

represented as

y u y = [ Y l , Y ? " " ? Yh-9 y,+[,'*',Yn]

y , = x,, r i s n , i # i a , jz,, = l , , . . . , c ,

v = 1 , 2 ; . . , g

Y .? = x,,, e , ,, (mod mi , , ) , 0 < e,,b< m,,??

l ~ i ~ ~ s k , a =. 2 . .. c

y,,, = xJ, ej l (mod m,&,),< e J J m,", + 1 I uI ,

(9)= I , 2 ; . . , g

where i , , i2 ; . ., ,. are the positions of errors in the infor-

mation digits, the corresponding error values being e,,,



20 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-11: ANALOG AN D DIGITAL SIGNAL PROCESSING, VOL.. 39, N O . I , JANUARY 1992

e;*, a , eic espectively; and j , , j , , . . . , are the positionsof errors in the parity, the corresponding error values being

number Y can be represented as

and

c

(mod mk+,.), + r # j ,,,, e,*,-. . . ejs, respectively. The altered information A , =

o , . . . , ~ , lc , , . . . , ~ ] (11) where r = 1 , 2 ; . * , n - k , an d v = 1, 2 ; * * ,g.

A fundamental property of the syndrome digits is stated in

where the following theorem.

Theorem 5: Under the assumption that no more than 0residues are in error, for the RRNS with d = h + /3 + 1

( h< 0) one of the following four cases occurs.

Case I : If all the syndromes A I , . , A n - are zero, then

no residue is in error, and vice versa,

For the following cases, let p be the number of nonzero

syndromes.

X - [ X I , X , , " ' > Xk]

E tf [ O ; * e , 0, e,, , 0; * * ,0, e , > ,0 ; * a , 0 , elc, , . . 0 ] .

Since E = 0 (mod m,)or all i # i,, Q! = 1, 2; * a , c, it is

a

an d m,'. herefore, E is an integer of the type

Of all information except m,l , m ~ z , ''

M

mi,miz mjcE = e'

Case 2: If p I , then exactly p corresponding parity

Case 3: If X + 1 I 5 0, then more than h residue

digits are in error, and all other residue digits are correct.

( 12 ) digits are in error .3 ia(mod mi=)

Case 4: If p + 1 I I - k , then at least one of the

w h e r e 0 < e' < m ;,mi2 . . . m ; Based on [ y l , information digits is in error.

y,, . . , yk ], and the base extension (BEX) operation [2], the

purpose of which is to avoid processing large valued inte-

gers, the parity digits are recomputed to get

J ' L + r = Y(mod m k + r ) 3 = 3 2 * * * 3 n - k . ( I 3 )

proof: The Proofs of Cases 1, 2, and 4 are given in [61

and [141. Therefore, they are omitted here. For Case 3, since

k correct information residues uniquely determine the correct

redundant residues, it is clear that if h + 1 I 5 3 and all

the information residues are correct, then more than X parity-

residues are in error. Now, suppose that c information

residues are in error. We have c 2 1, i.e., at least onee define the test quantities A, called the synd romes as

A,. = yi+,. - y,+,.(mod mk+,), = 1 , 2 ; . . , n - k . information residue is in error. Define A> = yi+,. -

(14)

Since both X an d E in (10) are less than M , we consider

x,+,.(mod mk+,).Then, f rom (9) and ( l l ) , we have

X t t [ X I > ~ , ' ' ' ~k ? k + l ' " ' ? X n J

rtt Y , , ' 2 , " ' , Y k , Y ; + , ~ ' . * > Y ~ J

the following two cases.i) X + E < M . In this case, F - x ~ [ o ; . . , o , ~ , , , o ; . . , o , ~ , ~ , o ; . . , o ,- M e l C , 0 , - . . , 0 ,',, A ; ; . . , A ' , - , ] .

Y = X + E = X + e ' v (15) It is obvious that if the received redundant residue digiLyk+,.

is erroneous, i .e. , y k + ,# x k + , . , hen A > # A r . For Y > X

(the case for r < X can be analyzed in an analogous man-

ner), since M > r - X , from (7) we know that at least

(16) d - c = X + P + l - c o f A ' , , A ' , ; * . , A ' , - , a r e n o n z e r o .Therefore, we have the following three assertions.

1) If at most /3 - c redundant residues are in error, then

at least h + /3 + 1 - c - ( p - e ) = h + 1 syndromes

are nonzero.

2) If at most h + 1 - c redundant residues are in error,

then at least /3 syndromes are nonzero.

3) If at most h - c redundant residues are in error, thenat least + 1 syndromes are nonzero.

na=lm,,

an d

M

L l m r ,

M

A,. e- c (mod m k + r ) , + # j y

A,. e'- - e,+,.(mod rnk + , . ) 7 k + r = J , (17)%= 1 m, ,

where r = 1 , 2 ; * * , n - k , an d Y = 1, 2 ; * * , g .

ii) M 5 X + E < 2 M . In this case,

- MY = X + E - M = X + ? - M

k = l m f n Now, for Case 3, if h + 1 to p syndromes are nonzero

contradiction with the assertion 3). This gives the proof for

Case 3. This proves the theorem.

c and no more than h residue digits are in error, then there is a

(18)' - n m , ,



S U N A N D KRISHNA: CODING THEORY AP P ROACH- P ART 11 21

If Case 2 of Theorem 5 takes place, the new parity digit

y i + r s the correct value of the erroneous parity digit y k + r .

If Case 4 of Theorem 5 takes place, a procedure to determine

error locations and error values for singleldouble errors is

described in the following sections. This procedure is based

on an efficient search and computation of the error location

and

M

m j

A , E (e?,.) - m,) -(mod rnk+ ' ) , r = 1 , 2 , . . , n - k

( 2 6 )

and their values from the given set of syndromes for the

specified error correction/detection capability of the code.This procedure will give rise to a unique solution if and only

if there is a one-to-one correspondence between the errors

that take place and the associated syndromes. In this regard,

necessary and sufficient conditions (as well as simplified

sufficient conditions) are described, which the moduli consti-

tuting the RRNS must satisfy for this procedure to work. In

the following sections, these conditions are described for the

specific case being analyzed. One of the reviewers presented

a proof of the sufficient condition for the above mentioned

to obtain the values e:"" and where r = 1 , 2 , . ., n

- k . Compare to check if the solutions to ( 2 5 ) or (26) ar e

,:2.2) = . . = ,!,n - k l . If either of the two conditions is

satisfied, for J = I, then the Ith information digit is declared

to be in error, the value of the error e, = e ( M / m , ) ( m o d

e, , , a n de = ,I) if e(2.1) = ( 2 . 2 ) = . . . = e f . n - k ) . The correct

value of the Ith residue digit is ( y ,- e,) mod m, , and y is

decoded to the codevector f , where

e(I.1)= (1.21 = . . . = e y l r e(2.11 =e, J

m,), where e = , if e:l.l) = (1.2) = . . = e ( 1 3 n - k ) .

J el

-,

x, = y J , = 1 , 2 ; . . , n , j # 1 (27)

(28)

one-to-one correspondence for the general case of h random

error co rrection and /3 random error detection in the review.

This proof is included in Appendix A. -,x,= y, - e , (mod m, ) .

IV. A PROCEDUREOR SINGLE-ERRORORRECTI ONND

MULTIPLE-ERRORETECTIONd = h + + 1 , h = 1 ,P > A)

By Theorem 4, the RRNS with d = h + /3 + 1 , h = 1 ,

an d fi > h can simultaneously correct single residue error

and detect P ( P = d - 2 ) errors. Assume that only the pth

information digit is in error; then ( 1 5 ) - ( 2 0 ) become the

following.

i) X + E < M . In this case,

- MY = X + e'-

m,

and

M

mPA e'-( m o d m , + , ) , r = 1 , 2 ; - . , n - k . (22)

ii ) M 5 X + E < 2 M . In this case,

It is clear from (21), ( 2 2 ) , and (25) that if one error takes

place in the pth information digit such that X + E < M ,

and I = p. Similarly, if one error takes place in the pth

information digit such that M I + E < 2 M , then for

e,from (23), (24), and ( 2 6 ) . However , it remains to be shown

that if the following two cases occur: 1) only one error takes

place in the pth information digit and j # p ; and 2 ) more

than one but less than d - 1 errors take place, then at leasttwo of {e:'.r); = 1, 2 ; - . , - k } and at least two of

{ r = 1, 2 ; * . n - k } are unequal, i .e. , (25) and (26)do not give consistent solutions. Under an additional con-

straint on the form of the moduli, the result is established in

the following theorem.

Theorem 6: If the moduli of an ( n , k ) RRNS code, are

such that there do not exist integers n,, n,; 0 I , < m,,

0 I c < M , = n:= m, e , 1 I , 5 k , 1 5 c 5 3 that sat-

isfy

then for ; p , e = e' = e ( l . l ) - (1,2)- . . . = e:l .n-klJ -

j ~ p , e = e' = = (2.21 = . . . = e ? , n - k ) and I =

c+ I

i = I(23) n,Mc + n,m, = n , , k < r, 5 n , 1 5 I (29)

and then for either of two cases, that is, Case 1 , only the pthinformation digit is received in error and J # p ; and Case 2 ,

more than one (I 6 ) esidue digits are received in error, the

solutions to ( 2 5 ) and ( 2 6 ) are inconsistent.

Proof: Case 1: fo r X + E < M , an d J # p , i f the

M

m,A r ( e ' - m )-( m o d m k + r ) , r = 1 , 2 ; . . , n - k

(24) solutions to ( 2 5 ) are consistent, i.e., e: '. ') -

(21)-(24) a procedure to correct single error and detect the

('a . . --

e Y k ) e, then comparing (25) to ( 2 2 ) , we obtainwhere 0 < e' < mp. Given A I , A 2 ; . . , A n - k , based on

presence of up to d - 2 errors can be outlined as follows. - (em , - e'm,) = 0 mo d n k + r

that is, em p - e'm, is a multiple of IIFIFmk+r.This is not

( r = t

M

For J = 1, 2 , . . , k , solve the cong ruences m P m J

M

A,e(1-r)- ( m o d m k + r ) , r = 1 , 2 ; - . , n - k ( 2 5 ) possible as 1 emp - e'm, 1 < II;::mk+r. Similarly, for

J+ E < M , and J # p, if = (2.2) = . - =J



22 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-11: ANALOG AN D DIGITAL SIGNAL PROCESSING. VO L 39. NO . I . JANUARY 1992

ey , n -k )= e, then comparing (26) to (22) we obtain

This is not possible as (e'm, + (m , - e)m,,) < 2mpm, <n,=, k+,. The case for M I + E < 2 M , can be ana -

lyzed in an analogous manner.Case 2 : fo r X + E < M ,

e Y k J e, then comparing ( 2 5 ) to ( I 6) we obtain

n - k

if e:'." = e:'.') = . . -

where k < r ,I , v = 1 , 2 , . . g. This congruence cannot

ho l d as l e I I ~ = , m I m e 'm, I < m , I I : ~ = , m , ~ y

m , . For example, assume that the pth information

digit and the first g parity digits are in error, and 0 c + g ,

where c = 1 , then the above congruence is

n8+1 g,= ,

M- ( emp - e'm,) = O(mod m,- ,m, , )

mJmP

This is not possible as 1 emp - elm, I < m n - m n . n other

words, at least two of { e:'.r); r = 1 , . . . n - k ) are un-equal. Similarly, for X + E < M , if e:2." = = . . .= e;2.n-k) = e, compar ing ( 2 6 ) to (16), we obtain

n - k - g

where k < r ,I , v = 1 , 2;. e , g. The above congruence

cannot hold as ( e l m j + ( m i - e)13',=,mi,t)< 2 m j

n : = l m i < v< Elf=+l'-grnr,,or c + g < 0; nd (29) for c + g

= 0. he case for M 5 X + E < 2 M ca n also be analyzed

in an analogous manner. This proves the theorem.

Lemma I : If the moduli of RRNS satisfy the condition

min (m, ,mr2}> max { 2 m i , m i 2 m i , - m, , ] ( 3 0 )

where k < r l , r 2 I n and 1 5 i , , i , 5 k , then for either of

the two cases, that is, Case 1 , only the pth information digit

is received in error and j # p ; and Case 2 , more than one

residue digit is received in error, the solutions of either (25)

or (26) are inconsistent.

Proof: The condition is sufficient since, if the moduli

satisfy (30), he n (29) is satisfied trivially. This com pletes the

proof.

It is interesting to note that this sufficient condition on the

moduli for single error correction and multiple error detec-

tion is the same as the sufficient condition on the moduli for

single error correction [14]. Based on the concept developed

above, and under the assumption that the moduli rnl, i = 1 ,

2 , . * , n , satisfy the condition in Theorem 6 or Lemma 1, the

algorithm to correct a single error and simultaneously detect

presence of multiple error in RRNS can be described asfollows.

Step I :

Step 2:

Step 3:

Step 4:

Step 5:

Step 6:

According to the received vector, we compute the

syndromes.

Check how many syndromes are zeros:

1) If all the syndromes are zero, then no erroroccurs. Stop.

2 ) If only one syndrome is nonzero, then only

one error in corresponding parity. Correct i t

and stop.

3 ) If d - 3 to 1 syndromes are zero, then go to

Step 6.

4) If all the syndromes are nonzero, then go to

Step 3.

Le t j = 1.

Perform single-error consistency-checking for the

nonredundant moduli m,. Check the consistency

of the solutions. If it is consistent, then go to Step

5 ; if it is in consistent, then j = j + 1. Go to step

4 fo r j I . Fo r j = k + 1 , go to Step 6.Only one error in the j t h position. Correct it and

stop.

Declare more than one error detected and stop.

A flowchart for this algorithm is given in Fig. 1.

Example I : Consider a (10, 6) RRNS code based on the

moduli m, 23 , m2 = 25, m3 = 27 , m4 = 29 , ms = 31,

m6 = 3 2 , m, = 67, m8 = 71 . m, = 73, m,, = 79, where

m,, m8. ,, and m,, are the redundant moduli. Clearly,

m,m8 > 2m,m, - m5 - m,, therefore, the condition in

Theorem 6 and Lemma 1 is satisfied. M = n;=,n , = 446 ,

623, 200. Le t X = 400, 000, 000. then x = [8, 0, 22, 13,

2.5, 0, 17, 58, 4 , 1 I ] . Since 0 I < M , x is a codevector.

Assume that one error takes place in the first residue digit,

and the received vector is [O , 0, 22 , 13 , 25 , 0, 17, 58, 4 , 1 I ] .

Based on the information part [O , 0, 22, 13, 2.5, 01 and BEX

operation, we compute the syndrome digits A , E 14(mod

67) , A , E 48(mod 71) , A 3 = 50(mod 73) , A 4 = 17(mod

79). Following the decoding algorithm, we check the consis-

tency for j = 1 , 2 , . . k . k = 6 in this example. Based on

computing e:"'" and r = 1 , 2 , 3 , 4, it is seen that

consistent solution to (26). So, declare that one error oc-curred in the first residue digit, the value of the error being

e , = e(M/m,)(mod m , ) = 15 . From (28), the estimated

value of the erroneous digit is given by jc? = 8.

Now, assume that two errors take place in the first and

third residue digits, and the received vector is [O , 0, 23, 13 ,

25 , 0, 17, 58, 4, 111. Based on the information part [O , 0,23, 13, 25, 01 and BE X operation, we compute the syndrome

digits A , = 23(mod 67), A 2 = 50(mod 71), A 3 = 63(mod

73) , and A 4 = 6S(mod 79). Following the decoding algo-

rithm, we check the consistency for i = 1, 2;.., 6 , it is

found that there is no consistent solution, so declare more

than one error detected.

when j = 1 , e ( 2 . 1 ) - e ( 2 . 2 ) - e ( ? . 3 ) - = e = 14 is aI - -



S U N A N D KRISHNA: CODING THEORY APPROACH-PART 11 23

A CCORDI NG TO THE

RE CE I V E D V E CTOR.

C O M P U T E T H E

S Y N D R O M E S

d = Atfi+l

d L 4 , h = l , P > h

Note Under the assumpt ion

I CONS I S TE NCY -CHE CK 1 NG

A T T H E N O N R E D U N D A N T

MODULUS ml

CHE CK HOW M A N Y OF

A N D S T O P

A RE DE TE CTE DS T O P

Fig. 1. A decoder flowchart for single-error correction an d multiple-errordetection.

V . A PROCEDUREOR DOUBLE- ERRORORRECTI ONVD

M ULTI PLE- ERRORETECTIONd = h + 0 1, h = 2 ,

P > A)

If Case 4 of Theorem 5 happens, i .e. , exactly two residue

digits are received in error, there are two possible cases; that

is, Case a): both errors in information part; Case b): one

error in information part and the other in parity part. By

Theorem 4, the RRNS wi th d = h + 0 + 1, h = 2 , and

P > h can simultaneously correct two errors and detect Perrors. N ote that for correcting two errors o nly, i .e . , for

d = 5 , the analysis of the error correcting procedure is the

same as in this section.Consider Case a): if the p th and qth information digits are

received in error, (15)-(20) become the following.


- MY = X + E = X + e-

mPm4(31)

and

ii) M 5 X + < 2 M . In this case,

- M MY = X + e- - M = X + (e ' - m P m q) _-

mPm4 mPmQ(33)

M

A , = (e ' - m,m,)-mod m k + r ) ,

mPm4

where 0 < e' < mPm4.Similar to the analysis in Section

IV, given A , , A , ; . . , A n P k , based on (31)-(34) a proce-

dure to determine error locations and error values can be

outlined as follows.

For i = 1 , 2 ; * . , k - 1, j = i + 1, i + 2 ; * * , , solve

the congruences

and

r = 1 , 2 ; . . , n - k (36)

to get the values

(37)

where r = 1, 2 , 3, 4. Based on {e". ' ) , e ( i ~ 2 ) } n d { e ( 2 3 ' ) ,

and by the CRT, we can write

where

Then check if the following two conditions are satisfied: i)

el:) < m,m, and (35) holds by substituting el;.) f o r e ( ' , r ) ,

i.e., the solutions to (35) are consistent, and the consistent

solution is e!)); ii) e;:) < mim, and (36) holds by substitut-

and the consistent solution is e$). Note that { e ( ' *3 ) ,e(',4)}

and { e",')} will be used for obtain ing the consiste nt

solution for Case b). Let us say either of the two conditions is

satisfied for i = f an d j = h. Then the fth and hth informa-

tion digits are declared to be in error, the value of errors

being

ing e::) fo r e"," , i.e., the solutions to (36) are consistent,

M

e = e-(mod m f ) (43)m J m h

.f -

Meh = e-(mod m h )

m f m h

(44)

= 2 7 ' ' 9 - (34) where e = e:;), if e:).' is the consistent solution to (35);



24 IEEE TRANS ACTION S ON CIRCUITS AND SYSTEMS-11: ANALOG AN D DIGITAL SIGNAL PROCESSING, VOL. 39, NO . 1 , JANUARY 1992

e = e!;), f ej;) is the consistent solution to (36). Note that if

e = O(mod m,) o r e E O(mod m,), then only the hth or ft h

information digit is in error. The correct values of the fth

and hth residue digits are yf - ef(m od m,) and y , -

e,(mod m,), respectively, and y is decoded to the codevec-

to r x, where

/,

x j = y ,, j = 1 , 2 ; . . , n , j f f , h

x, = y , - ef (mod m,)

(45)

(46)

(47)

A

Ax h = y h - e , (mod m, ) .

It is clear from (31), ( 3 2 ) , and (35) that if two errors take

place in the p th and 9t h information digits such that X + E

< M , hen for i = p and j = q , e = e' = e:;),f = p , and

h = q . Similarly, if two errors take place in the p th and qth

information digits such that M I X + E < 2 M , th en f o r

i = p an d j = q, e = e' = e!;), f = p , and h = q from

(33), (34), and (36). However, it remains to be shown that

the solutions to (35) and (36) cannot be consistent, if any one

of the following cases occurs: 1) only one error takes place in

th e p t h ( p # i, j ) nformation digit, 2) exactly two errorstake place in the p th and qt h information digits, where

p # i, j and / o r q # i, j , 3) exactly two erro rs occur: o ne in

the information part and the other in the parity part, 4) more

than two (less than 0 1) errors occur. Under an additional

constraint on the form of the moduli, the result is established

in the following theorem.

Theorem 7: If the moduli of an ( n , k) RRNS code are

such that there do not exist integers n,,, n,; 0 5 n,, <m , m , , 0 I n, < M e = II:=,rnlu, 1 I , I k , 1I I ;

that satisfy

C f 2

i = 1

nijMe + n , m i m j = n m , , k < ri I n , 1 I , j I k

(48)

then for any one of the following three cases: 1) only one

error takes p lace in the p th ( p # i, j ) nformation digit, 2)

exactly two errors take place in the pth and qth digits,

where p # i, j and/or q # i, j , 3) more than two (less than

0 1) errors occur, the solutions to (35) and (36) are

inconsistent.

The proof of this theorem is similar to the proof of

Theorem 6, and therefore, is omitted here. The sufficient

condition for Th eorem 7 can be shown to be exactly the same

as the sufficient condition stated in Lemma 1.

Now, consider Case b): assume that one error takes place

in the pth information digit and the other takes place in the

( k + u)th digit (the uth parity digit); then (15)-(20) become

the following.i) For X + E < M , r = 1 , 2 ; - . , n - k , and 1 5 U I

n - k ,

- MY = X + E = X + e'-

m P(49)

an d

M

mPA = e'-( m od m k + r ) , r # U

ii) For M r X + E < 2 M , r = 1 , 2 ; . . , n - k , an d

I s u r n - k ,

and

M

mPA , = ( e' - m p ) - - ek +u (mo d mk +u ) (54)

where 0 < e' < m p, and m 4 can be any one of the informa-

tion moduli. Based on (49)-(54), the procedure to determine

error locations and error values is the same as that for Case

a), except that when no consistent solution is found in Case

a), then we check if any n - k - 1 congruences in either

(35) or (36) have a consistent solution, say e, which satisfies

e < m i m j a nd e E O(mod m,) or (mod mj). Note that this

consistent solution may be obtained from the combination

{ or { e(2 ,3) ,e(2,4)),ince one error may occur inthe ( k + 1)th or ( k + 2)th digit. Let us say the above

condition is satisfied for i = f and j = h, and the only

inconsistent congruence is corresponding to the ( k + s)thresidue digit, then the fth (if e = O(mod m,)) and ( k + s)th

residue digits are declared to be in error. The correct value

of the fth and hth residue digit can be obtained from (43)

and (44), and the correct value of the ( k + s)th residue digit

is as follows:

/-.. Mx k + s E k + $ + A s - m k + s ) (")

m f m h

if n - k - 1 congruences in ( 3 5 ) have a consistent solution

e ; and

if n - k - 1 congruences in (36) have a consistent solution

e. It is clear from (49)-(51) and (35) that if one error takes

place in the pth information digit and the other takes place in

the ( k + u)th digit such that X + E < M , then for i or

j = p , say i = p , e = = e 'm, = O(mod m,), f = p ,



SUN AND KRISHNA: CODING THEORY APPROACH-PART 11 2s

an d s = U , where r = 1, 2 ; . . , n - k , an d r # U . Simi-

larly, from (52)-(54) and (36) if one erro r takes place in the

pth information digit and the other takes place in the ( k +u)th digit such that M I + E < 2 M , t he n f or i or j = p ,

say i = p , e = e(2")= elmj = O(mod mi), f = p , an d s

= U , where r = 1 , 2 ; - . , n - k , an d r # U . However, it

remains to be shown that neither (35) nor (36) has exactly

n - k - 1 consistent solutions that are equivalent to zero

mod mi or mod m j , if any one of the following cases

occurs: i) only one error takes place in the pth ( p # i , j )

information digit, ii) exactly two errors take place in the pth

and qth information digit ( p # i , j , a n d/ or q # i , j ) , ii)

exactly two errors take place in the pth and ( k + u)th digit

( p # i , j ) , iv) more than two (I /3) errors take place.Under the same constraint on the form of the moduli as stated

in Theorem 7, the above result is established in the following

it is consistent, then go to Step 6; if it is inconsis-

tent, then go to Step 7.

Step 6: If the consistent solution e = O(mod mi ) or e =O(mod m j ) ) , hen only one error in the jth (or

ith) position; otherwise, exactly two errors in the

ith and jth positions. Correct them and stop.

Step 7 If exactly d - 2 solutions are consistent and the

consistent solution e = O(mod m,) (o r e = O(mod

m j ) ) , hen exactly two errors: one in the jth (orith) position and the other in the parity for which

the solution is inconsistent. Correct them and

stop. Otherwise, go to Step 8.

Step 8: j = j + , go to Step 5 fo r j I . For j > ,i = i + 1 go to Step 4 for i 5 k - 1. For i = k ,

go to Step 9.

Step 9: Declare more than two errors detected and stop.

theorem.

Theorem 8: If the moduli of an ( n , k ) RRNS code a re

such that there do not exist integers n i j , n,; 0 5 n i j <mimj , 0 I , < M , = I I :=,rnjm, 1 I , i k , 1 I I 3that satisfy

A flowchart for this algorithm is given in Fig. 2 .

Example 2: Consider the same RRNS as in Example 1,

based on the moduli m , = 23 , m2 = 25, m3 = 27 , m4 =

29, ms = 31, m6 = 32, m7 = 67, m, = 7 1 , m, = 7 3 , m,,

= 79, where m7, ma, mQ, nd m, , are the redundant, ._

c +2 moduli. Clearly, m7m, > 2m,m6 - m 5 - m,; therefore,

the condition in Theorems 7 and 8 is satisfied. M = rI ;=,rn,= 446, 623, 200. Let X = 400, 000, 000, then x = [8, 0,

22, 13, 25, 0, 17, 58, 4, 111. Since 0 5 X < M , x is a

n l J M c+ n,m,m, = n m , , k < T I 5 n , 1 5 i , j II = 1

(57)

then when (35) or (36) has exactly n - k - 1 consistent

solutions equivalent to zero mod m j or mod mi , he ith or

jth information digit and the parity digit corresponding to the

only inconsistent solution are received in error.


Theorem 6, and therefore is omitted here.

Based on the concept developed above, and under the

assumption that the moduli of MDS-RRNS satisfy the neces-

sary and sufficient condition in Theorem 7 or 8, or the

sufficient condition in Lemma 1, the algorithm to correct

double-error and simultaneously detect multiple errors can be

described as follows.

Step 1: According to the received vector, we compute the

Step 2: Check how many syndromes are zero:

syndromes.

1) If all the syndromes are zero, then no error

occurs . Stop.

2) If only one syndrome is nonzero, then only

one error in corresponding parity. Correct it

and s top.

3) If two syndromes are nonzero, then two errors

in parity. Correct them and stop.

4) If d - 4 to 2 syndromes are zero, then go to

Step 9.5) If none or one syndrome is zero, then go to

Step 3.

Step 3: i = 1.

Step 4: j = i + 1.

Step 5: Perform double-error consistency-checking for all

the combination of the nonredundant moduli mi

an d mj. Check the consistency of the solutions. If

codevector. Assume that two errors take place in the first and

third residue digits, and the received vector is [0, 0, 23 , 13 ,25, 0, 17 , 58, 4 , 111. Based on the information part [0, 0,

23, 13 , 25 , 01 and BEX operation, we compute the syndrome

digits A , = 23(mod 67), A , = 50(mod 71), A 3 = 63(mod

73), and A 4 = 65(mod 79). Following the decoding algo-

rithm , we check the consistency for i = 1, 2; * a , 5 , j = i +1 , i + 2 , ,6, based on computing {e".'), e(',,), e(13 3),

from (37) and (38) and

{el;) , l;)} rom (39) and (40). It shows that when i = 1 an d

j = 3, er:) = 217( < m l m 3= 621) is a consistent solution to

(36), and e!:) f O(mod 23) or (mod 27). So, declare twoerrors occur in the first and third residue digits, and from

(43)-(47) the estimated values of the erroneous digits are

{ e(*,'),

+.I = y , - e(2)-M (mod m , ) = 813 m l m 3

Now, assume that two errors take place in the first and

( k + 4)th residue digits, and the received vector is [0, 0, 22,

13, 25, 0, 17, 58, 4, 121. Based on the information part [0,

0, 22, 13, 25, 01 and BEX operation, we compute the

syndrome digits A , = 14(mod 67), A , = 48(mod 71), A 3 =

50(mod 73) , A 4 = 16(mod 79). Following the decoding algo-rithm, we check the consistency for i = 1 , 2 , . . . 5 , j = i +1 , i + 2 , . . . 6 , based on computing {e'' . '), e(',,), e(19 3),

from (37) and (38) and

{ell.), l:)} from (39) and (40). It shows that when i = 1 and

j = 2 , e!:) = 35 0 = O(mod m,) < m,m, is a consistent so-

lution to the first three congruences of (36), i.e., the only

{



26 IEEE TRANSACTIONS ON CIRCUITS A N D SYSTEMS- U: A N A I . 0 G AND DIGITAL SIGNAL PROCESSING. VOL 39. NO. I . JANUARY 1992

d = A+P+I

ACCO RDING TO TH E d 25. A=2, P>h

R ~ ~ ~ ~ ~ ~ ~ f ~ ~ R ~ote Under the assumptionthat no more than d-3

res idues ar e i n e r r o rS Y N D R O M E S

T W O E R R O R SNO ERROR

STOPH E C K

HOW MANY

P E R F O R M T H E D O U B L E - E R R O R

CO NSISTEN CY- CHECKIN G FO R THE

COMBINATION OF mi AND m,C H EC K H O W M b N V S O L U T l O N S I R E

C O N S I S T E N T

EXACTL Y TW O

EXACTL Y TW O ERRO RS O NE

i t h A N D j t h it h POSlTlON

PO SIT IO N

C O R R E C T T H E MCO RRECT THEMTOP

STOP.

I , - , + I I

Denotee=O(mod m,) 1 = 1 + 1 4 Iy lelmi-o, where

e is the consistent ERRO RS A RE

MORE T H A N TWO

D E l E C T E D S T O Ps o l u t i o n

Fig. 2 . A decoder flowchart for double-error correction an d multiple-errordetection.

inconsistent congruence is corresponding to the modulus

ml0 .So, declare that two errors occur in the first and tenth

residue digits, the estimated values of erroneous digits being

given by (46) and (56), that is,

/\ Mx , E , - e!;)- (mod m , ) = 8

mlm2

,’\ Mx , ~y,, + A 4 + ( m , m ,- e!<’)-- (mod m,”) 11

mlm2

Note that i is always equal to 1 , if two errors occur in a wa y

that one error is in information part and one error is in parity

part.

V I . A P ROCEDURE F OR S INGLE-BURS T E R R O R

CORRECTION

A single-burst residue error vector of length b is defined

as an RNS vector whose nonzero residue digits are confined

to b consecutive digits, the first and last of which must be

nonzero. The decoding procedure for performing burst-error

correction depend s on the criterion of performance. If residue

errors occur in burst such that

P(burst error of length a ) < P(burst error of length b )

a > b, and the criterion of performance is to minimize the

probability of decoding error (maximum likelihood decoding

(MLD)), then the decoding procedure can be summarized as

follows. Decode y to a codevector x for which the length of

burst-error vector e, e = y - f , s minimum. We will use

this decoding procedure in our subsequent analysis. Here

P(A) denotes the probability of occurrence of event A. T he

proof of Theorem 9 is similar to the proof of lemma 3 in

[ 141, and therefore, is omitted here.

Theorem 9: For detecting all single-burst errors of length

b or less , an RRNS code must have d >_ b + 1.

Theorem IO : For correcting all single-burst errors of

length b or less, an RRNS code must have d 2 2 b

+1 .

Proof: Let x, be a codevector other than x in !d. The

Hamming distance among x , x , , and y satisfies the triangu-

lar inequality

d ( Y , x) + d (Y , x , ) 2 d ( x , X I ) .

Since d ( x , x, ) 2 d an d d ( y . x) = w t ( e )5 b , ( e = y -

X ) ,

d ( y , x ) 2 d - b .

If d 2 26 + 1 , then d ( y . x, ) 2 b + 1 . thereby implying

that the length of burst-error vector e is smaller than the

length of any other burst-error vector e , , e , = y - x,. On

the other hand, if the codevector x is such that wt( ) = d,

and the nonzero digits of x are confined to d consecutive

places (examples of two such codevectors are codevectors

corresponding to the integers X , = n:l,’m, an d X , =

2nf;=-,’ in, ,espectively), then for d 5 2 6 , we can show that

there exists at least one codevector x, such that d ( y , x ) =

w t ( e ) 2 w t ( e , ) . e , = y - x , . In this case, incorrect decod-

ing will take place. This proves the theorem.

The proof of‘ Theorem 11 is similar to the proof of

Theorem I O , and therefore. is omitted here.

Theorern 11: For correcting all single-burst errors of

length b or less and simultaneously detecting all burst errors

of length b’ ( b ’ z b) or less, an RRNS code must have

d r b + b ’ + l .

A fundamental property of the syndrome digits is stated in

the following theorem.

Theorem 12: For an MDS-RRNS ( n , k ) code, ( d = n -

k + 1 = b + b‘ + l ) , under the assumption that no morethan one single-burst residue error of length 5 b’ occurs,

on e of the following four cases occurs.

Case I : If al l the syndromes A , , A 2 , . , an^ are zero,

then no residue is in error, and vice versa.

For the following cases, let p be the number of nonzero

syndromes.

Case 2: If p 5 b syndromes are nonzero, then exactly p

corresponding parity digits are in error, and all other residue

digits are correct.

Case 3: If b + 1 5 p 5 6‘ yndromes are nonzero, then

more than b residue digits are in error .

Case 4: If 6‘ 1 5 p 5 n - k syndromes are nonzero,

then at least one of the information digits is in error, and all

the last b + 1 redundant residue digits are correct.

The proof of this theorem is similar to the proof ofTheorem 5 , and therefore. is omitted here. If Case 4 of

Theorcm 12 happens, and exactly one single-burst residue

error of length I occurs, there are two possible cases,

that is, Case a) : all the errors are in the information part;

Case b) : the errors are in both information and parity part.



SUN AND KRISHNA: CODING THEORY APPROACH-PART I1 27

Consider Case a). Assume that a single-burst error of

length c (1 I I ) akes place between the ith and ( i + c

- 1)th information digits, i.e., in the position interval [ i ,

i + c - 11, then (15)-(20) becom e the followin g.


M

Y = X + E = X + e ' ( 5 8 )ne= m;+a-

and

A r = e'M

(mod m k + r ) , = 1 , 2 ; * - , n - k .'= Imi+a- 1

(59)

ii) M I + E < 2M . In this case,

a consistent solution to (63). In other words, check if e:') <nfi=mi+a- and (62) holds by substituting e:') fo r or

e?) < IIZ=,m,+,- I , and (63) holds by substituting e?) fo r

e?,r). Let us say either of the two conditions is satisfied for

j = p. Then declare that a single-burst error of length Ioccurs in the position interval [ p , p + b - I]. The values

of the errors are

where e = e'') if e$') is the consistent solution to (62);

e = e? ) if e$) is the consistent solution to (63). The correctM values of the erroneous residue digits are ( y , - e,) mo d

m , , w h e r e q = p , p + l ; * * , p + b - l . T h e n y i s d e -

coded to the codevector x, here

x q = y q , q = 1 , 2 ; . . , n , q f p , p + l ; * * , p + b - 1A

(60)

-Y = X + E - M = X +

an d

A

x, = y q - e,(mod m , ) , q = p , p + l ; . . , p + b - 1.

(mod m k + r ) j

(70)

r = 1 , 2 ; . . , n - k (61)

where 0 < e' < n:= l m i + a- l .Given A , , A , ; . . , A n -k ,

based on (58)-(61) a proced ure to determine erro r locations

and error values can be outlined as follows.

For j = 1 , 2 , . . ., k - b + 1, solve the congruences

r = 1 , 2 ; * . , n - k (62)

r = 1 , 2 ; . - , n - k (63)

to get the values and e?,'). Using the last b digits of

{ ejl,r);= 1 , 2 , * e , n - k } and the last b digits of {

r = 1, 2; * * , n - k } , and by the CRT we compute the

following values:

n - k

r = n - k - b + 1

n - k

e:) = ey*r)7' 'Mi mod M i ) (64)

e?) e?.')T;M; (mod M i ) (65)r = n - k - b + 1

where

M iM ' = -

mk + r

T M ; = l ( m o d m k + r ) (68)

and then check if ejl:' is a consistent solution to (62) or is

Note that e , = O(mod m,) implies that no error occurs in

the qth digit. It is clear from (58), (59), and (62) that if a

single-burst error of length c ( c 5 b) takes place in the

position interval [i , i + c - 11 such that X + E < M , he n

fo r [ j , j + b - I] including [ i , i + c - 11, e' m = e:'),where m is the product of the moduli corresponding to the

position interval [ p , p + b - 13 exclusive of [ , i + c - 11.

Note that if c = b, then e' = e$')and p = i. Similarly, if a

single-burst error of length c ( cI ) takes place in the

position interval [ i , i + c - 11 such that M I + E <2 M , then for [ j , j + b - 11, including [ i , i + c - 11,

e'm = e?) from (60), (61), and (63). For this case, if

c = 6 , hen e' = e?) and p = i. However, it remains to be

shown that neither (62) nor (63) has a consistent solution, if a

single-burst error of length c (cI ' ) takes place in the

position interval [i , i + c - 11 which is not included in [ j ,

j + b - 11. Under an additional constraint on the form of the

moduli, the result is established in the following theorem.

Theorem 13: If the moduli of an ( n , k ) RRNS code a re

such that there do not exist integers n B , n,; 0 I B < M B

1I I '; that satisfy

- nR=lm,e, o I , < M , = n z = l m i a , s j a , i, < k ,

c+ b

I = 1

nBMc + ncMB = n mr, ,k < r ,I (71)

then for [ i , i + c - 11 not included in [ j , j + b - 11,neither (64) or (65) has a consistent solution, where a

single-burst error of length c (1 I I ' ) occurs in the

position interval [i, i + c - 11 .

Proof: Consider the extrem e case, that is, [ , i + c - 11

an d [ j , + b - 11 do not overlap and c = b' . For X + E

< M , if (62) has a consistent solution e$' ) , hen comparing



28 IEEE TRANSACT IONS ON CIRCUITS AND SYSTEMS-11: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 39, NO . I , JANUARY 1992

(62) to (59) we get

b b'

e' II m j + , - l - el1)IIi a = 1 a = l

where e' < II%= , m i + , - This cannot hold due tob b'

e' II m j + a - l- e;.')II m i + a - li a = l a =

b 6' n - k

a = a = 1 r = I< n m j + o l - l m r + u - l < II m k + r .

Similarly, for X + E < M , f (63) has a consistent solution

e?), then comparing (63) to (59) we get

b'where e' < n a = , m i + a - l .his cannot hold due to (71). The

case for M I + E < 2M , can also be analyzed in an

analogous man ner. This proves the theorem.

A sufficient condition for T heorem 13 can be shown

to be exactly the same as the sufficient condition stated in

Lemma 1.

Now, consider Case b): In Case a), when j = k - b + 1

if we do not find the consistent solution, it implies that a

burst error of length > b occurs or a burst error of length

5 b occurs in both the information and parity digits. Assume

that a single-burst error of length c (1 I I ) akes place

in the position interval [ k - c 1 + 1 , k + c2] ,where c l ,

c22 1 and c1 + c2 5 b, then (15)-(20) beco me the follow-

ing.


an d

M

A e' (mod m k + r ) ,:= k - ,+ i

r = c2 + I ; . . , n - k . (74)

ii) M 5 X + E < 2 M . In this case ,

F = X + E - M= X

r = c2 + l ; . . , n - k (77)

where e' < II:= k - c l + m . A procedure to determine error

locations and error values is described in the following.

For j = k - b + 1, check if there exists the largest num-

ber c, in the range [I, b ) such that either of the following

two conditions is satisfied.

Condition 1): Th e last (b ' + c,) consecutive congruences

of ( 6 2 ) have a consistent solution, i.e..

and

Condition 2 ) : The last (b ' + c3) consecutive congruences

of (63) have a consistent solution, i.e.,

an d

Le t us say there exists a number c , such that either of the

above two conditions is satisfied; then declare that a single-

burst error of length I takes place in the position interval

[ k - c3 + 1, k + b - c 3 ] .The estimated values of the er-

rors and erroneous information digits can be obtained by (69)

an d (70) fo r p = k - b + 1. The inconsistent congruences

correspond to the erroneous parity digits, y k + s , s = 1,

2 , . . , b - c 3 , he estimated values of erroneo us parity digits

being:



S U N AN D KRISHNA: CODING THEORY APPROACH-PART I1 29

if e:’) is the consistent solution to ( 6 2 ) ;an d

M

(mod (79)‘:=lmk-b+a

if e?’ is the consistent solution to ( 6 3 ) .

It is clear from (72)-(74) and ( 6 2 ) that if a single-bursterror of length c (1 5 c 5 6) takes place in the position

interval [ k - c, + 1 , k + c,] uch that X + E < M , where

c , , c2 L 1 an d c , + c2 = c , then for j = k - b + 1 , ( 6 2 )

ha s 6’ + b - c, solutions, which are consistent and equiva-

len t to ze ro mod m k - b + o i , i . e . , = e$’’ =

e n , = , m k - b + a , w h er e CY = 1, 2 ; * e , b - C , . r = c, + 1,

c, + 2 ; * * , n - k , b - c2 = c 3 , an d c, = c4. Similarly,

from (75)-(77) and ( 6 3 ) , if a single-burst error of length c

(1 5 c I ) akes place in the position interval [k - cl + 1 ,

k + c,] such that M I X + E < 2 M , where c , , c2 2 1

an d C , + c, = c , then for j = k - b + 1, ( 6 3 ) ha s b’ + b

- c, solutions, which are consistent and equivalent to zero

m od m k - b + a , i .e ., e?..) = e?’ = e n , = , m k - b + a , w he re

c2 = c 3 , and c , = c4. However, it remains to be shown thatno c3 and c, exist to satisfy Condition 1) and 2 ) , if a

single-burst error of length > b (but I ’) occurs or asingle-burst error of length I occurs not in the position

interval [k - b + 2 , k + b - 11. The result is established

in the following theorem.

Theorem 14: If the moduli of ( n , k ) RRNS code are such

that there do not exist integers n B , n,; 0 I B < M B =

I I t z I r n j m ,0 5 n , < M , = H:=lmicy , 1 I a , i , < k , 1 Ic I ’; that satisfy

b - c ,

b - c ,

CY = 1 , 2 ; - * , b - c,, r = c2 + 1 , c, + 2 ; * . , n - k , b -

c + b

;= I

n B M c + ncMB = n , , k < r ,I (80)

then when a single-burst error of length > b (but I ’)

occurs or a single-burst error of length I occurs not in the

position interval [ k - b + 2, k + b - 11, c3 an d c4 do not

exist to satisfy Conditions 1) an d 2 ) .


Theorem 13, and therefore, is omitted here. Based on the

concept developed above and under the assumption that the

moduli satisfy the condition in Theorem 13 or the sufficient

condition in Lemma 1, the algorithm to correct single-burst

error of length b or less and simultaneously detect the

presence of single-burst error of length b’(b’ > b) in RRNS

can be described as follows.

Step 1: According to the received vector, we compute the

Step 2: Check how many syndromes are zero.

syndromes.

1) If all the syndromes are zero, then no error

occurs . Stop.

2 ) If p (1 I 5 b ) syndromes are nonzero, then

exactly p corresponding parity digits are in

error. Correct them and stop.

3) If b + 1 to b‘ syndromes are nonzero, then go

to Step 8.

Step 3:Step 4:

Step 5:

Step 6:

Step 7:

Step 8:

4) If 6’ + I to all syndromes are nonzero, then

Let j = I .

Perform b-error consistency-checking for the

combination of the nonredundant moduli m,,m j+ , . . . m j s b - ,. Check the consistency of the

solutions. If it is consistent, then go to Step 5; if

it is inconsistent, then for j < k - b + 1 go toStep 6 , go to Step 7 for j = k - b + 1.

A single-burst error in the position interval [ j ,

j + b - 11, correct it and stop.

j = j + 1, go to Step 4.

If the last b’ + c consecutive congruences of ( 6 2 )

o r ( 6 3 )have a consistent solution which is equiva-

lent to zero mod rIi:C mk- b+a, then a single-

burst error exists in the position interval [ k - c

+ 1 , k + b - c]. Correct it and stop. Otherwise

go to Step 8.

Declare that a burst-error of length > b is de-

tected. Stop.

go to Step 3.

A flowchart for this algorithm is given in Fig. 3 .

VII. E XT E NS IONSF PREVIOUSLGORITHMS

In [4] and [8]-[lo], the algorithms for locating a single

residue digit error are based on the properties of modulus

projection and MRC. MRC is an operation to represent the

integer X in the form

fl 1 - 1

I = ] r = lx = a , r I m r (81)

w h e r e O I a , < m , , I = 1 , 2 ; . . , n , a n d n ; = , m , = 1 . In

an ( n , k ) RRNS code, the first k mixed radix digits a,,

a 2 , a , ak are called the nonredundant mixed radix digits,

and the rest ak+ a k + , , . . , a, will be called the redundant

mixed radix digits. It is obvious that if the redundant mixed

radix digits are all zero, the number X is a legitimatenumber; otherwise, X is an illegitimate number. T he modu-

lus m, - projection of X in an ( n , k ) RRNS code, denoted

by X , , is defined as

X , = X [mod- M m T ]

that is, X , can be represented as [x, , Z;’ , xi-,,

x,, , a , x,] which is the residue representation of X in areduced RRNS with the ith residue digit xi eleted. The

mixed radix representation of Xi s

fl 1 - 1

r = lxi = aln m r

I # ; r # i

where the new first k mixed radix digits are still called the

nonredundant mixed radix digits and the rest are called the

redundant mixed radix digits , e .g. , if i 5 k , a , ,

a 2 , . ai- , ai+ ,. . a k , ak+ , are the nonredundant

mixed radix d igi ts , and a k+ * , a k f 3 , - ., a, are the redun-

dant mixed radix digits. It is obvious that if X, s a legiti-



30 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-11: ANALOG AN D DIGITAL SIGNAL PROCESSING, VOL. 39, NO . 1 , JANUARY 1992

i . e . ,=b+b tb ,bccording to the

received vectorcompute the k + c

r = I

Note Under the assumption hatno more than one single-

< b occursburst residue error of length M' = n ,, i , 5 k + c , i,,, > k + c , i , I

r@

k

r = 1 (86)' = M = n m,, i, > k .

It is obvious that M' 2 M . M' is also the smallest nonzero

number represented by (85) with all the nonredundant mixedradix digits zero. It follows from (84) that the MA-projection

of any legitimate number X in RRNS is still a legitimate

number in [0, M ) , that is XA= X . The proof of the

following theorem has been given in [4], and therefore, is

fo r the combinationof the nonredundant

Check how m a n y o f

.L

Y $? +IA burst residue error 1of length >btk-b+l

I s detected Stop IYL

9 7 Denote e=0 (mod mj ) by leimr=O.where e is the consistent solution

Fig. 3 . A decoder flowchart for single-burst erro r correction.

mate number, the redundant mixed radix digits are all zero.

However, if the redundant mixed radix digits are all zero,

then mathematically, X i could still be an illegitimate num-

ber. In [8], it was shown that an illegitimate projection

resulting from a single error cannot be smaller than the

smallest nonzero number represented by (83) with all the

nonredundant mixed radix digits zero. Therefore, Jenkins's

algor ithm [SI-[lo] for locating a single residue digit erro r

consists in checking whether the redundant mixed radix digits

are all zero for each modulus m,-projection, i = 1, 2 , . n.

In the following analysis, we will show, from the coding

theory point of view, that their algorithm can be extended for

detecting and correcting multiple residue digit errors. The

MA-pro jection of X , denoted by X,,, is defined by

XA= X mod-i 21 (84)

where M A = r Ik= ,mIm, = { i , , i2; . ., ,, . . , i,; i , < i,

< . . . < i ,< < i h } , an d hr - k = d - 1. X,

can also be represented as a reduced residue representation of

X with the residue digits x j l , i 2 , . , xi, deleted. Then, the

mixed radix representation of X , is

n 1 - 1

The legitimate and illegitimate range of the reduced RRNS

are [0, M ' ) an d [M', MM, /MA ) , espectively, where M'

is the product of the first k moduli of the reduced RRNS,

omitted here .

Theorem 15: Le t x be a number in RRNS and TA e

the MA-projection of 2, where MA = IIk=,mIu.f TA#

x,he residue representation of x, uniquely differs from xin on e or more of the residue digits corresponding to the

moduli mi , ,m,?; . , mlA.

Before considering the range of the MA-projection 2, f

any illegitimate number x n RRNS, we give the following

theorems. Theorem 16 is based on the fact that all codevec-

tors differ in at least d places in an RRNS with minimumdistance d.

Theorem 16: If X is a legitimate number in the RRNS

having minimum distance d, then any integer x difiering

from X in at least one and no more than d - 1 residue

digits is an illegitimate number.

Theorem 17: Le t 2 be an illegitimate number in the

RRNS. If there exists a legitimate number X differing from

x in the i,th, i,th; * e , i,th residue digits, then the M A -projection X, is a legitimate number, where MA=

Proof: Since 2 differs from X in the i , th,

., , th residue digits, x an be expressed as follows:

n L l m I u .

i , th,

where 0 < e' < nL=,mjm.y definition,

= x, < M .

Therefore, FA s a legitimate number. This proves the

theorem.

Theorem 18 : Let x be an illegitimate number in the

RRNS having minimum distance d. If the MA-projection

TA,M A = I I ~ = , m l u ,I < d ) , is a legitimate number,

than there exists only one legitimate number X differing

from x n one or more of the residue digits corresponding to

the moduli m,,, m r 2 , a , m,,.



SU N AND KRISHNA: CODING THEORY APPROACH-PART I1 31

Proof: Since x # x,,t is obvious from Theorem 15

that the legitimate number X = x, s a solution to the

problem. Now assume that there exist two different legitimate

numbers X nd X ' , both differing from 2 n on e or more

of the residue digits corresponding to the moduli m ,,,

is affected by p or fewer errors corresponding to the moduli

m,,, ,? , . . . m],,.rom the definition of the reduced RRNS,

i . e . , (86). x., can be expressed as follows:

MMRd

,, X,, e' .

m i z , - . . , T h e n , ~ m = , ~ l , , ~ c y = l m , , ,

where 0 5 e, e' < n:=lm, ,x . Since 0 5 X, ' < M , 0IX , = X ' , < M , a n d M M R / I l k ; l m , ~ x M , it follows that

e = e' = 0 an d X = X' . This contradicts the original as-

sumption X # X' . This proves the theorem.

Theorem 19: In an RRNS with minimum distance d = h

+ p + 1, an illegitimate number x is originated from a

legitimate number X affected by h errors corresponding to

the moduli m,,, m, , , . a , m lA , f an only if the M.,-pl_qjection

FAs a legitimate number and the M,-projection X, s an

illegitimate number, where M A = FIk= l m l , , i M , =n : = l mj o , 1 5 P 5 3, a n d g c d ( M p . M.,) f n c y = l m l , t

Proof: By The orem 17, the M,-projection J7', is-

legitimate number. Since gcd(M,, M,) # n k = l m l , r ,X,

can be treated as a number originating from X, affected by

h or fewer errors corresponding to the moduli m , , ,m i Z ; ~ ~ ,,,, that is,

M*.I

where 0 < e' < r I : = l m, cx . By Theor em 1, M , 2

FI:=lmjar12= lm ,c y, t follows that xp s an illegitimate

number. This proves the necessity. The sufficiency is proved

by contradiction. Since FAs a legitimate number, by Theo-rem 18 there exists only one legitimate number X differing

from x n one or more of residue digits corresponding to the

moduli m, , , m,?,. Lm,A. How ever, if the legitimate num -

be r X differs from X in less than X residue digits (say p

( p < A) residue digits corres pon ding to the moduli mL,m,,; a , mjp ) then, by T heorem 17, the Mp-projection X,

is a legitimate number, where M p = r Im l c z nd gcd( M,,

M,) # M,,. This contradicts the original assumption. This

proves the theorem.

The range of the illegitimate projection Fa, f any illegiti-

mate number x s given in the following theorem.

Theorem 20: Under the assumption that no more than /3

residue errors occur, for an RRNS with d = h + + 1 an d

P_> h, if the M,,-pojection x,, f any illegitimate number

X is illegitimate, X, will be in the range [ M', MM, /M.,),where M A = r I k z l m , , x , 1 I , 5 n , an d MI is the lower

bound of the illegitimate range of the reduced RRNS.

Proof: The illegitimate projection x, an be treated as

a number originating from X,,n the reduced RRNS which

' gc d ( a , b ) denotes the greatest common divisor of a an d b

. mod where e '< n mi,,a = I

MR

where i,. 5 k + c , i,.,, > k + c.

By Theorem 1 , M R 2 ~ ~ _ , m , + m n k = c + l m , ~ ~ n , = l m , , ~ ,i t follows that

x,X ,

+M' 2 M' . This proves the

theorem.

Therefore, by checking only the redundant mixed radix

digits, it can be determined whether or not the projection is

legitimate. If all the redundant mixed radix digits are zero,

the projection is legitimate; otherwise, it is illegitimate.

Based on the above analysis, a procedure for correcting

double errors and detecting multiple errors can be outlined as

follows.

Step 1: Based on the received residue vector, compute the

mixed radix digits and check if all the redundant

mixed radix digits are zero. If yes, then declare

no error and stop. O therwise, go to next step.

Step 2: Compute the mixed radix digits of the m,mj-

projection for i = 1 , 2 ; . ., n , j = 1 , 2; e . , n ,an d i # j . Check if all the redundant m ixed radix

digits are zero. If yes, then declare the ith and

j t h residue digits are received in error, correct

them by base extension, and stop. Otherwise, go

to next step.

S t e p 3: Declare more than two errors detected. Stop.

VIII . DISCUSSION

The present-day practical algorithms for residue-error c or-

rection and detection mostly focus on single-error correction

because of the considerations of large memory space require-

ment or computational inefficiency fo r multiple error correc-

tion. Our algorithms developed above do not require large

memory space as required for table lookup. They also seem

to be much superior to the algorithms in [6] an d [9] f rom acomputational efficiency point of view, i.e., the requirement

of multiplications ( MU LT ) and additions (AD D) [13]. The

comparison of our algorithms derived in Sections IV and V

with those in [6] an d 191 in terms of the requirement of ADD

and MULT is shown in Table I. We should point out that for

double-error correction the expression in column 2 of Table I



32 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-11: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 39, NO . 1, JANUARY 1992

TABLE I

T H ER E Q U I R E M E N TF M U L T OR A D D FO R S I N G L E / D O U B L ER R O R S O R R E C T I O N~ ~ ~ ~~~~~~~~~

# Error Yau and Liu’s Jenluns’s NewCorrect ion Algori thm [6 ] Algorithm [9] Algori thm

1

d = 3

2

d = 5

$ k 3 + $ k 2 + ik - 2 ( M U L T )

$k3+ f k 2+ + k ( A D D )

i k 3 + : k 2 + 2 k ( M U L T )

$ k 3+ t k z + 2 k ( A D D )

$ k 4+ $ k 3 + Yk’ + T k + 8 ( M U L T )

ak4 + T k 3 + T k 2 + T k + 8(ADD)

i k ’ + j k - I ( M U L T )

i k 2 + i k + ] (ADD)

F k 2 + T k - 2(MULT)

$ k 2 - y k + 2(ADD)

No t applicable

is obtained by extending Jenkins’s algorithm [9] as described

in the previous section. For single-burst error correction, our

method only needs to find n - k syndromes, whereas Yau

and Liu’s method has to find n syndromes. The comparison

is shown in Table 11. Mandelbaum proposed a decoding

procedure for multiple-error correction based on continued

fraction expansion and Euclid’s algorithm [151- 171. Since

this procedure needs to process large valued integers and use

an iterative process, it appears to be more suitable for a

general purpose computer. Also, it is difficult to compare

Mandelbaum’s algorithms to the algorithms presented in this

paper as the approaches are entirely different.

APPENDIXPROOF F GENERALAULT-DETECTION,

FAULT-CORRECTIONASE

In this Appendix we give an alternate proof of conditions

that allow detection of up to ,6 faults and correction of up to

h faults in a maximally redundant RRNS. We follow this

with the general procedure for finding and correcting these

faults.

., , be the actual locations of the faults in the

information digits, and let j , . , , be the locations of the

faults in the parity digits, where c + g I . Le t I; be the

reconstruction of the value modulo M using only the infor-

mation digits, but let us define the error E differently than in

Section 111. In particular, define

Le t i, ,

E = F - X .

The value E can take one of M possible values. Without

computing F or X directly, however, we can definitely state

that E is confined to the range

-M < E < M .

Now let e, = E ( m o d m;).Clearly all the digits e , are zero

except for digits i = i , * e , i,. We can therefore state that E

can be written in the form:

M

where

Let us define syndromes in the sa me way as b efore. Then

it is easy to show that

(arbi t rary , e l se .

Le t 9 an d 9’ be any (possibly overlapping) sets of indicesin the range 1,. . .,k . Suppose 9 ha s c elements and 9’ ha s

c’ elements. Let = n a’ be the intersection, with

C” I in ( c , c’) elements. Let 9’ = CP - be the ele-

ments of that are not in a’, and let a * = a’ - a’,* e the

elements of 9’ that are not in a . Now we assume that

2 1 1 m , n m , I I m k - n m , - n m k < I I m , .E @ ’ ’ JEQ’ kc@’ J E Q ’ kcQ r e t

(A . 3 )

where \E is any subset of at least c + c’ of the redundant

moduli k + 1, * , n . Note that this assumption is trivially

satisfied if

2m,m, < m,ms (A . 4 )

for any 1 I , j 5 k an d k + 1 5 r , s 5 n .Now for the main result. Assume (A.3) holds, and assume

the RRNS is maximally redundant. Now hypothesize that the

syndromes we observe might have been caused by some

alternative “different” combination of up to 0 igit failures

in information digit positions i; , . , ‘,,, and parity digit

positions j ; , .,f,!, here c’ + g’ 5 0, and h + ,6 = n -

k . (“Diffe rent” implies that at least one error informationdigit in the alternative hypothesis is assumed to have a

different value than in the correct set of error information

digits.) We show by contradiction that the syndromes of this

alternative hypothesis cannot equal those from the correct

hypothesis. U nder the alternative assumption, we hypothesize

an error - M < E’ < M , and can write this error in the

form:

ME =e’-

n:=lm,;

where

Le t @ = { i , , - * , i,] be the set of information digits thatactually failed. Let CP‘ = { i ; , . i;,} be the set of informa-

tion digits assumed to have failed in the alternative hypothe-

sis. Let 9 = { j , , . . , } be the set of parity digits that have

actually failed. Let P’ { j ; , * a , jb.}e the set of parity

digits assumed to have failed in the alternative hypothesis.



SUN AND KRISHNA: CODING THEORY A PPROACH-PART I1 33

TABLE I1

T H ER E Q U I R E M E N TF M U L T OR A D D FOR SINGLE B U R S T RROR ORRECTION

Yau and Liu’s Algorithm [6] New Algorithm

k 2 + ( 4 b 2 - 3 b ) k - 4 b 2

k 2 + ( 4 b ’ - b ) k ( A D D )

k - (5b’ - 4 b ) ( M U L T )

1 1 2 b - 5- k 2 + ~ k - (46’ - 7 6 + 2 ) ( A D D )

2 1 2 2

k = 8 2 268 (MULT)b = 4 2 2 9 4 ( A D D )

188 (MULT)166 (ADD)

k = 12

b = 4 2 648 (ADD)

2 616 (MULT) 338 (MULT)292 (ADD)

k = 16 2 1280 (MULT)b = 8 2 1336 (ADD)

728 (MULT)

654 (ADD)

Le t e; = E’(mod m,) e the error digits in the alternative

hypothesis, with e: # 0 for all i E W . Note that this implies

that e’(mod mi)+ 0 fo r i E a’. Define

a n a‘I.2 =

@ I 7 @ - $’ J

a2 = @I’@ 1 , 2

9’ lp - q 1 . 2

9 2 = 9‘ q 1 . 2 .

= @ I 9 2 + @ I

@’ = @ I 9 2 + r p

= 9 1 . 2 + l p l

9‘ = 9 1 . 2 + 9 2 .

9 1 . 2 = 9 n 9’

Note that

Now if the alternative hypothesis could indeed explain the

observed syndromes, then we would have

Equating this with the actual value of the syndromes (A.2) in

terms of e, and using our index set definitions, we get

which implies that

1[(enmi - e’n m i )EO 2 re@’ n €* I * mn €* I min €* 2 m,

(mod n m,) = 0 .

r e { k+ 1;. , } ‘?I - I - q 2

But since the moduli are relatively prime, this implies that

(en m i - e’n m,) 6 n mrI€ @ i €@l r€ {k + 1 ; . n } - k’ ’-*I - q 2

for some integer 6. Multiply both sides by the product of th e

redundant moduli in index sets

e n m, - e ’ n m i )

9’,nd q 2 o get

n rn , = 6MR. ( A . 6 )ic@’ re*‘ . ’ +I‘ k2

No w

e n m , - e’ n m ,i€@ I

= 2 n m , - n m , - n m,., € @ I + @ * + @ i€** I€*’

Suppose set @’,’ has e” elements, where c” 5 min ( e , e ’ ),and suppose set has g ” elements , where g ” I in ( g ,

g ’ ) . By our assumption (A.3), the expression above is guar-

anteed to be less than the product of any set of c + e‘ or

more redundant moduli. Note that the set + 9 ’ 9*

ha s g + g’ - g ” redundant moduli in it, which leaves X + p- ( g + g’ - g ” ) 2 c + e’ + g” redundant moduli not in

that set. Choose any c

+e’ of these, and call this set

q.Then

en m, - e’n mi mi-

< n m , I R .

Combining this with (A.6) shows that the constant 6 = 0,

and we have

re% + I. ’+ +q 2

en m, - e ‘ n m , ( A . 7 )i€ @

But this is only possible if

e = t?n , , e‘ = m ,i€ Q ic*’

which would imply that

Also, this implies that e, = 0 for all i E @ ’ , and that e; = 0

for all i E a 2 .



34 IEEE TRANSACTIONS O N CIRCUITS AND SYSTtMS-I1 ANALOG AND DIGITAI. SIGNAL PRO CI-SI NG. VOL 39. NO. I . JANUARY 1992

Now, we originally assumed that there were nonzero er-

rors in all the index positions in the set +, including those in

the set 9 ' .We m ust conclude that 9 ' must be a null set. and

so 9 = Our alternative hypothesis assumed that there

were nonzero errors in all the index positions in the set +',including those in the set + I . We must conclude that if the

syndromes match, then 9' is a null set, and so a ' =

Putting this together, 9 = a', and so if the syndromes match

then the alternative hypothesis must have correctly identifiedthe faulty information digits. Also note that (A.7) implies that

if 9 ' an d (P2 are null, then e = e' an d E = E' . Thus the

error information digits e, an d e ; must also match, and the

alternative hypothesis exactly matches the actual errors. We

conclude that with Assumption (A.3) and with up to X faults.

the syndromes uniquely specify the erroneous information

error digits.

A procedure to correct up to X rrors is as follows: Check

the number of nonzero syndromes. If there are none, then no

errors occu rred. If there are fewer than A, hen all errors are

in the parity digits. If there are greater than X but fewer than

/3 errors, then m ore than X rrors occurred. If there are 0 r

more nonzero syndromes, then consider all possible sets of

zero up toX

errors in information digits. For each hypothe-sis, try to solve ( A .2 ) fo r a consistent value e in the range

(A.1) . At most one of these combinations can yield a consis-

tent solution e (either positive or negative) which would

explain the non-faulty syndromes. If none yield a consistent

solution e, there must be more than X faults. Total complex-

ity is the time required to compute the base extension, plus

the time to evaluate every possible hypothesis involving X

failures.

ACKNOWLEDGMENT

The authors wish to express their thanks and appreciation

towards the reviewers for their thoughtfulness, thoroughness,

and for providing constructive criticism and insight into

many aspects of the error control problem in R R N S . W e

have a greatly improved and complete paper due to theirefforts and time. The associate editor is also to be thanked for

his assistance and guidance during the review process. The

proof in Appendix A is taken from the review of one of the

reviewers.

REFERENCES

N . S. Szabo and R. I. Tanaka. Residue Arithmetic and i ts Applica-t ion to Computer Technology.

R . W. Watson and C. W. Hastings. "Self-checked computation using

residue arithmetic." Proc. IEEE. pp . 1920- 1931, Dec. 1966.

D . Mandelbaum. "Error correction in residue arithmetic," IEEETrans . Cornput . , pp. 538-545, June 1972.

F. Barsi and P. Maeutrini , "Error correcting properties of redundantresidue number system$." IEEE Trans . Cornput.. pp. 307-315.Mar. 1973.

-. "Error detection and correction by product codes in residuenumber system." I € € € Tr a ns . Co m p u t . . vol. C-23 , pp , 915-924 ,Sept. 1974.

S . S- S Yau and Y-C Liu. "Error correction in redundant residuenumber systems." I E E E Trans . Cornput . . vol. C-22. pp. 5- 1 I , Jan.1973.

V . Ramachandran. "Single residue error correction in residue numbersystems." IE€€ Trans. Cornput.. pp . 504-507. May 1983.

M . H . Etzel and W . K . Jenkins. "Redundant residue number systemsfor error detection and correction in digital filters," I € € € Trans.Acousi .. Speech. Signal Processing, pp . 538-544, Oct. 1980.

W . K . Jenkins and E. J. Altman. "Self-checking properties of residue

number error checkers based on mixed radix conversion." I E E ETrans. Circuits Sysr . . pp . 159-167. Feb. 1988.

W . K . Jenkins. "The design of error checkers for self-checkingresidue number arithmetic." / E € € Trans. Conipui.. pp. 388-396.Apr. 1983.

M. J . Bell , Jr. and W . K . Jcnkin,. "A residue to mixed radix

convener and error checker for a five-moduli residue number system."in Proc. IE€€ I n t . Conf. A S S P . vol. 3 . 1984.

C:C. Su and H.-Y. Lo . "A n algorithm for scaling and single reaidueerror correction in residue number systems." IEEE Trans . Conzpu t ..

A. P . Shenoy and R. Kumaresan. "Fast base extension using aredundant modulus in RNS." I€€E Trans . Cornput . , pp. 292-297,Feb . 1989.

H . Krishna. K. Y. Lin. and J L D . S un. "A coding theory approach toerror control in redundant residue number systems-Part I : Theoryand \ingle error correction." pp. 8- 17. this issue.

D. M. Mandelhaum. "On a class ot' arithmetic codes and a decodingalgorithm." / € € E 7rans . In form Theory . vol. IT-22, pp. 85-88.Jan. 1976.

-. "Further results o n decoding arithmetic residue codes." I€€€

Trans In form. Theory . vo l . IT-24. pp. 643-644. Sept. 1978.

-. "A n approach to an arithmetic analog of Berlekamp's algo-ri thm." / E € € Trans. Inforr??. Theor y. vol . IT-30, pp . 758-762,

Sept. 1984.

New York: McCraw-Hill , 1967.

pp. 1053-1064. Aug 1990.

Jenn-Dong Sun. for a photograph and biogra phy, please se e page 17 of thisi \ w e .

Hari Kriahna. for a photogrdph and biography. please see page 17 of thisi \ \ue

sun krishna ii

Documents