ieee transactions on information theory, vol. 55, no. 9, september 2009...

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 9, SEPTEMBER 2009 4155

Single-Exclusion Number and the StoppingRedundancy of MDS Codes

Junsheng Han, Member, IEEE, Paul H. Siegel, Fellow, IEEE, and Ron M. Roth, Fellow, IEEE

Abstract—For a linear block code �, its stopping redundancy isdefined as the smallest number of check nodes in a Tanner graphfor �, such that there exist no stopping sets of size smaller thanthe minimum distance of �� Schwartz and Vardy conjectured thatthe stopping redundancy of a maximum-distance separable (MDS)code should only depend on its length and minimum distance.

We define the �� -single-exclusion number, �� as thesmallest number of �-subsets of an �-set, such that for each�-subset of the �-set, � � �� , there exists a �-subset thatcontains all but one element of the �-subset. New upper boundson the single-exclusion number are obtained via probabilisticmethods, recurrent inequalities, as well as explicit constructions.The new bounds are used to better understand the stoppingredundancy of MDS codes. In particular, it is shown that for�� MDS codes, as ��, the stopping redun-dancy is asymptotic to �� , if � � �

��, or if � � �

��,

��, thus giving partial confirmation of the Schwartz–Vardyconjecture in the asymptotic sense.

Index Terms—Erasure channel, iterative decoding, maximum-distance separable (MDS) code, single-exclusion number, stoppingredundancy, stopping set, Turán number.

I. INTRODUCTION

T HE stopping redundancy of a linear code characterizesthe minimum “complexity” (number of check nodes) re-

quired in a Tanner graph for the code, such that iterative erasuredecoding achieves performance comparable to (up to a constantfactor, asymptotically) maximum-likelihood (ML) decoding. Itcan be viewed as a basic measure of the complexity–perfor-mance tradeoff in the use of redundant parity checks (RPCs)in an iterative decoder on the erasure channel.

Although this tradeoff is less straightforward to understandin non-erasure channels, there is empirical evidence that RPCscan improve performance in belief-propagation decoding on anadditive white Gaussian noise (AWGN) channel [1], [2], and, re-

Manuscript received December 07, 2007. Current version published Au-gust 19, 2009. This work was supported in part by the Center for MagneticRecording Research at the University of California, San Diego (UCSD), bythe National Science Foundation (NSF) under Grant CCF-0829865, and by theUnited States–Israel Binational Science Foundation (BSF), Jerusalem, Israel,under Grant 2002197. The material in this paper was presented in part at theIEEE International Symposium on Information Theory, Nice, France, June2007.

J. Han was with the Department of Electrical and Computer Engineering,University of California, San Diego, La Jolla, CA, USA. He is now with Qual-comm Inc., San Diego, CA 92121 USA (e-mail: [email protected]).

P. H. Siegel is with the Department of Electrical and Computer Engineering,University of California, San Diego, La Jolla, CA 92093-0401 USA (e-mail:[email protected]).

R. M. Roth is with the Computer Science Department, Technion–Israel Insti-tute of Technology, Technion City, Haifa 32000, Israel (e-mail: [email protected]).

Communicated by T. J. Richardson, Associate Editor for Coding Theory.Digital Object Identifier 10.1109/TIT.2009.2025578

cently, the concept of stopping redundancy has provided partialmotivation for novel decoding algorithms that achieve near-MLperformance for short, high-rate codes [3].

Formally, we define stopping redundancy as follows. Let bean linear code, and let be a parity-checkmatrix for . We shall assume that , but maybe larger than . The Tanner graph is a bipartitegraph with variable nodes, each corresponding to one columnof , and check nodes, each corresponding to one row of ,such that variable node is adjacent to check node if and onlyif . A stopping set in is a set of variable nodes suchthat all check nodes adjacent to the set are connected to the set atleast twice. It is well known [4] that iterative erasure decodingis successful if and only if the set of erasure locations does notcontain a stopping set. The size of a smallest nonempty stoppingset, referred to as the stopping distance and denoted by ,is therefore an important parameter governing the performanceof the iterative decoder. It is clear that , and it is notdifficult to see that equality can be achieved for any code, forexample by choosing the rows of to be the nonzero codewordsof the dual code . This leads to the following definition.

Definition 1: Let be a linear code with minimum distance .The stopping redundancy of , denoted by , is the smallestinteger such that there exists a parity-check matrix for with

rows, and .

Stopping redundancy was introduced by Schwartz and Vardy[5], [6], and was further studied in [7], [8]. The concept waslater extended in a number of interesting ways [9]–[11]. Relatedconcepts, such as stopping set enumerator, and generic erasure-correcting sets, were studied in [12], [13], and in [14], [15],respectively.

In this paper, we study the stopping redundancy of maximum-distance separable (MDS) codes. A code is MDS if it satis-fies the Singleton bound [16] with equality. Hence, anlinear code is MDS if . MDS codes are in-teresting for their unique properties and their close relation toother problems in mathematics and engineering. They are alsovery widely used in practice, a prime example being the class ofReed–Solomon codes [17].

For the rest of the paper, unless otherwise noted, denotesan linear MDS code. In [6], it was shownthat for all

(1)

where

0018-9448/$26.00 © 2009 IEEE

Authorized licensed use limited to: Univ of Calif San Diego. Downloaded on October 23, 2009 at 14:18 from IEEE Xplore. Restrictions apply.

4156 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 9, SEPTEMBER 2009

is the minimum distance of , the dual code of . The authorsof [6] then made an intriguing conjecture that should infact be a function of just and .

Note that the upper bound in (1) is never better than

So the upper and lower bounds can differ by up to a factorof . In [8], Han and Siegel observed that the upper boundcan be improved by introducing a new combinatorial quantity,the single-exclusion number, which we describe below. Beforedoing so, we first review two related, well-studied combina-torial constructs. For positive integers , and an

-set1 , an -Turán system [18] is a collection of -sub-sets of , called blocks, such that each -subset of the -setcontains at least one block. The -Turán number, de-noted hereafter by , is the smallest number of blocksin an -Turán system. A concept “dual” to that of a Turánsystem is that of a covering design [19]. Specifically, for

and an -set , an -covering design is a collectionof -subsets of , also called blocks, such that each -subset ofthe -set is contained in at least one block. The -cov-ering number, denoted by , is the smallest number ofblocks in an -covering design. Clearly, by definition

The stopping redundancy of an MDS code is closely relatedto covering/Turán numbers. In fact, the lower bound in (1) wasshown by noting that to maximize , the supports of min-imum-weight rows of must form an

-covering design (equivalently, the complements of supportsform an -Turán system). Hence

We now define the single-exclusion number, which was in-troduced in [20].

Definition 2: For an -set and , an -single-ex-clusion (SE) system is a collection of -subsets of , calledblocks, such that for each -subset of , ,there exists at least one block that contains all but one elementfrom the -subset. The SE number, , is the smallestnumber of blocks in an -SE system.

Note that an -SE system is a special kind of-Turán system. Note also that if we require all rows in the

parity-check matrix to be of minimum weight, thenis equivalent to the condition that the sets of column indicescorresponding to the zeros in each row form an -SEsystem. Hence

Therefore, any upper bound on is an upper boundon .

In [8], a number of upper bounds on were obtainedusing combinatorial constructions and were shown to be su-perior to the upper bound in (1). It was shown that

for , , andfor . It was also shown that

1An �-set is a set that contains � elements. Similarly, if � is any set, then a�-subset of � is a subset of � that contains � elements.

is asymptotic to (and to ) asfor any fixed , and that it is asymptotically at most

for any fixed .In this paper, we build upon the work in [8] and investigate

through a number of different approaches. New upperbounds are obtained and analyzed. They are then used to showthat as , is asymptotic to if ,or if .2 Hence, in an asymptotic sense, theSchwartz–Vardy conjecture is proved in these cases. For all

, it is shown that . A lower bound onis also derived.

The connection between SE systems and the stopping redun-dancy of MDS codes provides ample motivation to better un-derstand their properties. In addition, they hold intrinsic appealas well-defined mathematical objects with close relationshipsto other combinatorial constructs, such as covering designs andTurán systems, that have found wide use in discrete mathematicsand its applications.

The rest of the paper is arranged as follows. Section II isdevoted to upper bounds on . The bounds are obtainedthrough three approaches: combinatorial construction, proba-bilistic methods, and recurrent inequalities. Certain asymptoticproperties of our bounds are observed and discussed, some con-firming the Schwartz–Vardy conjecture in an asymptotic sense.The upper bounds are also compared numerically with the bestbounds found for up to . In Section III, we derive a lowerbound on and observe its implications. Section IV fo-cuses on the Schwartz–Vardy conjecture. We prove the conjec-ture for all MDS codes with , and comment on why

may be a reasonable estimate of . Section Vconcludes the paper.

II. UPPER BOUNDS ON

We start with some preliminaries. For any set , let de-note the set of all -subsets of . We say that covers anotherset if . Hence, if is an -set, then isan -SE system if and only if for each and

, there exists a block in that covers . A coveringdesign/Turán system/SE system (with prescribed parameters) issaid to be minimal if it contains the least number of blocks.

By definition, an -SE system is also an -Turánsystem. Hence, we have

(2)

where the second inequality [22] follows by noting that eachblock in the Turán system is contained in distinct

-subsets.Let be an linear MDS code. Recall

that the stopping redundancy of is related to SE and Turánnumbers in the following way:

2We adopt the standard “�-notation” and related asymptotic expressions[22, Ch. 9]. Functions �� and �� are said to be asymptotic to each other,denoted by �� , if �� , or equivalently, if�� , where �� stands for any function that goesto zero as � goes to infinity. More generally, we write �� if��


HAN et al.: SINGLE-EXCLUSION NUMBER AND THE STOPPING REDUNDANCY OF MDS CODES 4157

or equivalently

We now focus on upper bounds on .

A. Probabilistic Bounds

Probabilistic methods were used to obtain general upperbounds on stopping redundancy [8], [23]. For the case ofSE number, we appeal to the same approach with a differentargument.

Let be an -set. Consider the following random experimentin which we build an -SE system, . In the firststep, for a prescribed real value , insert into eachelement of with probability . The expected size of atthis point is but some -subsets, , may notbe covered. The probability that a given -subset is not coveredequals where

So, as a second step, for each , , that isnot yet covered, insert into some element of that covers

. The expected size of is then bounded from above by

This implies the following upper bound on .

Theorem 1: For all ,

(3)

Alternatively, in the first step of the random experiment, wemay instead make random drawings from . At the end ofthe first step, the probability that a given -subset is not coveredequals

if the drawing is done with replacement, and equals

where , if the drawing is done without re-placement. The results are the following bounds.

Theorem 2: For all

(4)

Theorem 3: For all ,

(5)

Theorem 3 is clearly stronger than Theorem 2, and is closelyrelated to Theorem 1. In fact, one can show that

So the upper bound (5), when minimized over , is no greaterthan

Note that we have strategically allowed to take the valuein the above expression. On the other hand, letting in(3), we see that the minimum value of the upper bound (3) is

Now, suppose the minimum value of the above expression is ,achieved at . Then its value at is less than .Therefore, we conclude that the upper bound (5) (when mini-mized over ) is less than the upper bound (3) (when minimizedover ) plus two. In practice, the difference between the twobounds is very small, while the upper bound in (3) is usuallyeasier to compute.

The upper bound in (3) can be written as

(6)

The first term in (6) is minimized when takes the value

in which case (6) becomes

(7)

where

(8)

As , it can be shown (see Appendix A) that if, then the term corresponding to prevails in the

sum (8). Therefore



Plugging the above into (7), we conclude that as , thebound (7) is3

ifif

if

where is any constant. By the choice of , and the factthat for the st term prevails in (8), no othervalues of give asymptotically tighter bounds than the above.

For the case when , a different asymptotic anal-ysis (see Appendix B) shows that the bound (3), when mini-mized over , is for all such that

, and is, particularly, for all.

B. Constructive Bounds

Our first construction for SE systems is based on a Turánsystem construction due to Kim and Roush [24].

Construction A: Let be an -set, and be a positiveinteger. For a prescribed positive integer , partition intosubsets, , , as equally as possible. Thus,

, such that for all . We will calleach a bin. Define

For , let

where

Note that in the above definition of , the subscript isto be interpreted as , and we shall stick to thisconvention where applicable.

Proposition 1: For all and all , asgiven in Construction A is an -SE system.

Proof: We show that any , ,is covered by a block in . If for some , let

be selected such that andis minimized. Such exists since , whichimplies that Now let be chosen suchthat if then . Note that covers

. Note also that , and that the choice ofand ensures . Hence, .On the other hand, if for all , select one

element in each such intersection, say . Now,choose such that , and consider ,

. All these sets cover , and since

3We write �� , if �� , and similarly, �� ,if �� , as ��.

, the set contains con-secutive integers, one of which must be congruent to modulo. Hence, for all , there exists such that is covered by

.

Theorem 4: For all integers

Proof: From Proposition 1, for all

Noting that

where

and that is a partition of , we have

An alternative (slightly looser) form of the upper bound isgiven in the following theorem.

Theorem 5: For all integers

(9)

Proof: Note that

The rest of the proof is similar to that of Theorem 4.

Corollary 1: For all fixed , as

Proof: Theorem 5 applies provided that . If, let . From (9), we have



For , let . If is even, the above derivation isstill valid. If is odd, then there is one bin that contains a singleelement, while the remaining bins each contain twoelements. From the proof of Theorem 5, we have

Hence

For , the result has already been shown in [8].

Since

Corollary 1 also implies that for any fixed

confirming a conjecture made in [8]. Note that the asymptoticfactor of in the above inequality is sharp for , in whichcase , while .For , stronger results can be obtained using recurrentinequalities, as shall be discussed in the next section.

Construction A is also a construction for Turán systems. In-deed, it can be viewed as an improved version (i.e., one withfewer blocks) of the Turán system construction in [24].

Proposition 2: For all and all , as given in Construction1 is an -Turán system.

Proof: The proof is similar to that of Proposition 1.

Our second construction for SE systems is inspired by a con-struction for Turán systems due to Frankl and Rödl [25].

Construction B: Let be an -set, and be a positiveinteger. Let , , and , , bedefined as in Construction A. For , let

where

and

are the number of “empty” and “full” bins for , respectively.For , the constructed subset of is

where is constructed as follows. Fix an arbitrary total orderon . Let be an index set that satisfies

, and is minimal in the sense that all propersubsets of violate this condition. For each such and ,

, let contain the -set that consists of all elements from

bins , the smallest elements of ,and the smallest elements of .

Proposition 3: For all and all , as given in ConstructionB is an -SE system.

Proof: We show that any , , iscovered by a block in . If , note that all -subsets ofcan be written as , for some . Since

for all , by choosing from different binsthat intersects, we can make take ondifferent values. Since no two of these values differ by morethan , this also means that we can realize differentvalues for . Since only numbers in

are greater than , there exists suchthat , hence ,and it covers .

If , consider two cases. First, let us assume that thereexists , such that and . In this case,remove from an arbitrary element in , add in otherelements from using as few elements from as possible,and call the resulting -set . That is, ,for some and some that has aminimal number of elements from . Note that the choice of

ensures that . Sincefor all , by choosing , , from

different bins where possible, we can make take ondifferent values. This also means

that we can realize different values for .Since only numbers in are greaterthan , there exists such that

, hence , and it covers .Next, if no exists such that and , this

means that for all such that , we have .Figuratively, it means that consists of a number of full bins.Let be any bin that intersects. Let be its largestelement. Take , and add to it elements from bins that

does not intersect, one bin after another, from the smallestto the largest within each bin, until is augmented tocontain elements. By construction, the -subset thus obtainedis contained in .

Theorem 6: For all positive integers

where

Proof: From Proposition 3, for any prescribed

To further bound , first note that



where if , and otherwise. Similarly

Now, since each is contained in precisely’s, we have

Hence

Finally, note that contains -subsets foreach valid , which must satisfy and

. Therefore, .

C. Recurrent Bounds

We observe that an -SE system can be constructed froman -SE system and an -Turán system,as shown in the following lemma.

Lemma 1: For all

or, equivalently, for all

Proof: Let be an -set and be an arbitrary ele-ment. Let be a minimal -SEsystem, and be a minimal -Turánsystem. Define . Then is an

-SE system. Indeed, for all , ,if , then there exists such that

, which implies that ,i.e., is covered by a block in . The only cases left are when

, and when . In either case, iscovered by a block in .

Theorem 7: For all

(10)

or, equivalently, for all

(11)

Proof: Recursively apply Lemma 1.

Interesting results follow. When , (11) implies that, which is sharp. When , since

, (11) gives

which is asymptotically tighter than Corollary 1.Generally, since exact values of most Turán/covering num-

bers are not known, the upper bounds in Theorem 7 often cannotbe directly evaluated. To get a computable upper bound, one cansimply replace each Turán/covering number in the sum by an ex-plicit upper bound. We show several ways to do this. The firstone is based on a result by Erdös and Spencer [26].

Theorem 8: For all

Proof: In [26], it was shown that for all

(12)

Plugging (12) into (11), we obtain the claimed result aftersome algebraic manipulations.

The second computable upper bound is based on an upperbound on Turán numbers due to Sidorenko [18, Construction 4].

Theorem 9: For all and positive integers

where

and .Proof: Omitted.

A third way to obtain an explicit upper bound from Theorem 7is based on a construction of an -covering design dueto Kuzjurin [27], although we count blocks in a slightly differentmanner.

Lemma 2: For all positive integers

Proof: Let be an -set. Without loss of generality, let. Let

and

s.t.



For each , we can add one block to , suchthat . Hence, by adding no more than blocks to ,we construct an -covering design. Therefore, forall

Now note that for all , , we have. Therefore, for all there is at most one block

in such that ; on the other hand, for everythere are elements such that . Hence

We have

Therefore, there exists , such that

Plugging the bound in the preceding lemma into (11), we ob-tain the following theorem.

Theorem 10: For all

Proof: Omitted.

Corollary 2: For all

In particular, as , if and , then

Proof: Omitted.

Interestingly, a similar asymptotic result can be shown forwhen is relatively small. First, we note the following

theorem relating SE and Turán numbers.

Theorem 11: For all

Proof: We will use the fact [28] that

From Theorem 7, we have

In [8, Theorem 21], it was shown that is asymptoticto for any fixed as . Theorem 11 enablesus to extend this result to all .

Corollary 3: For all

In particular, as , if , then

Proof: Note that

The result then follows immediately from Theorem 11.

D. Comparison of Upper Bounds

We numerically computed several of the upper bounds on, and hence on , for all .

For each pair, the tightest bound is identified, and the re-sults are shown in Fig. 1. Other bounds in the comparison that donot appear in Fig. 1 are those of Theorem 8, [1, Theorem 27],[1, Theorem 39], and the upper bound in (1), which is due toSchwartz and Vardy [6]. Note that is not considered,since in this case (and ) is known to be at most

, for which either precise formulas areknown, or tighter special upper bounds exist [8].

We observe that Theorem 9 appears to be the best upperbound except when the code rate is low, in which case The-orems 1, 4, and 10 give better results. Theorem 6 yields thetightest bound for just a few occasions with small values ofand .

Some examples of the upper bounds are given in Table I. Thetightest are highlighted in boldface. For comparison, a lowerbound on (hence on and aswell) has been included, based on (13) (see Section III). Com-pared to upper bounds previously known, significant improve-ments are observed. As a side note, we caution that while The-orem 10 is an excellent bound when is very close to , it be-comes loose quickly as gets smaller, and should be avoided ifthe code rate is greater than .



Fig. 1. Best upper bounds on �� (hence on ��), for ��.

TABLE IUPPER BOUNDS ON �� AND ��

III. LOWER BOUNDS ON

A few other lower bounds on Turán/covering numbers areknown, besides the simple lower bound in (2). For example,Schönheim [28] showed that

(13)Another useful bound is due to de Caen [29]

Note that a lower bound on is in turn a lowerbound on and on .

In this section, we study a lower bound result on(which is not a lower bound on in general). Notethat a lower bound (just) on is not necessarily a lowerbound on .

Theorem 12: For all

Proof: Let be an -set, and be a minimal-SE system. For each , can be partitioned into

blocks that contain and those that do not, namely

where and .Note that for all , is an -Turán system. Further,if we let , then each member of

includes an element of as a subset. To seethis, suppose . Then is a -setand so there exists such that . Since

, we must have . Hence, and.

On the other hand, since each element of is contained in

-subsets of , it is contained in at most distinct-sets in . Therefore

This lower estimate can be improved by a more careful argumentas follows. Let

...

...

Note that for all . Since is finite, thereexists such that for all . Regardless, define

We claim that on average, an element in is contained in atmost -sets in . To see this, considera process in which we enumerate elements of , and for eachelement, “mark” the -sets in that contain it.We start with elements in and proceed to , andso on. Each element in is contained in -sets in

, at least one of which lies in . Therefore, for eachelement in , at most -sets are marked. Now, forany , by definition, there exists , such that

. Hence, among the -sets that contain , atleast one of them, namely, , is already marked. Therefore,processing any marks at most additional-sets in . A similar argument shows that among

the -sets that contain an element of , at least oneof them is already marked after elements of have beenprocessed.



For , we show that on average, each element marks atmost -sets in . Let . As

, there exists such that .Since , we have and, hence, .In fact, , since otherwise it would implythat . Now, let , and denote by thenumber of elements of that are contained in . Notethat , since contains both and . Therefore, ofthe elements that are contained in , each on average marks

-sets in . For other elements in , theabove argument can be repeated until all elements have beenconsidered.

Based on the preceding discussions, we conclude that on av-erage, each block in is contained in no more than

-sets in . Hence

(14)

Since each block of appears in of the ’s, we have

Summing (14) over all , dividing both sides by , and notingthat (since was chosen to be minimal) gives thedesired inequality.

Corollary 4: For all , we have

Proof: Simply use the facts that

and

Equivalently, letting , we have that for all

hence the following corollary.

Corollary 5: For all , as

Proof: Trivial.

For fixed , the value is asymptoticto (cf. [30]). So the above corollary shows that forany fixed , the ratiois bounded away from as .

IV. ON THE SCHWARTZ–VARDY CONJECTURE

Schwartz and Vardy [6] conjectured that the stopping redun-dancy of an MDS code only depends on its length and minimumdistance. Han and Siegel [8] showed that for , theupper and lower bounds on for any MDScode differ by at most . Using Theorem 7, we can now closethe gap completely and prove the Schwartz–Vardy conjecturefor all MDS codes with .

Theorem 13: For all

Proof: Let be a minimal -Turán system, .We show that must also be an -SE system. Since isan -Turán system, all -sets are covered. In [8], it wasshown that all -sets and -sets are covered too. It remains toshow that all -sets are covered. Suppose, to the contrary, thatsome -set, , is not covered. It was shown in [8] that at least

blocks in contain elements from , and those thatdo not form an -Turán system. Hence

However, from Theorem 7

And since (cf. [18])

we have

Together, these imply that

Putting the upper and lower bounds on together, we have

However, since it is known (cf. [31], [32]) that, the above inequality results in a contradiction

for all . For , the theorem can be verifieddirectly, and was also proved in [8] (for ) usinga different argument.

Summarizing Theorem 13 and results in [8], we reach thefollowing conclusion.

Proposition 4: If is an MDS code, and, then

Proof: For , it is easy to findthat , , , and verify that



in all three cases. Otherwise, since, it follows immediately from

Theorem 13 and [8, Theorems 14 and 16] that

in all other cases.

In addition, from Corollaries 2 and 3, we see that as ,if , or if and , then

So in these cases we may say that the Schwartz–Vardy conjec-ture holds in the asymptotic sense.

Our approach regarding the conjecture has been so far to showthat in some cases the upper and lower bounds on stopping re-dundancy (SE and Turán numbers, respectively) converge, ei-ther exactly or asymptotically. However, we have seen that inother cases the corresponding SE and Turán numbers can beprovably different, even in the asymptotic sense (for example,see the discussion following Corollary 5), which shows the lim-itation of the current approach in fully resolving the conjecture.

It was conjectured in [20] that for an MDScode

(15)

the proof of which would in turn prove the Schwartz–Vardy con-jecture. We have shown that (15) is true for andthat it is true in an asymptotic sense for , and for

, . A reasonable question to ask is: what ifboth and are larger than ? For example, what ifapproaches a constant? The current approach only boundsto within a factor of up to . For example, using the result ofTheorem 8, we have

while, for the lower bound, we saw that

Alternatively, let us make the following observation. Supposewe are given one optimal parity-check matrix, i.e., one withrows that maximizes stopping distance. It is not apparent that allrows should have minimum weight, but suppose rows are ofminimum weight and the rest are not. We can replace each rowthat is not of minimum weight (and whose weight is, of course,at most ) with no more than minimum-weightrows, such that the union of supports of these rows is preciselythe support of the row they replaced. It is simple to verify thatthe replacement procedure does not decrease the stopping dis-tance, which also implies that the rank of the matrix is not re-duced. After all rows that are not of minimum weight have beenreplaced, we obtain a parity-check matrix with at most

rows, all having minimum weight, that achieves maximum stop-ping distance. Therefore

Now note that

so we have

Without knowing better how compares with, if we just ignore the second term, we obtain

This shows that in many cases, is a good estimate of. For example, if the code rate ,

then

And, clearly, for any constant code rate, is within a constantfactor of .

V. CONCLUDING REMARKS

While we have obtained a fairly good understanding of theSE number, and in some cases come close to uncovering the truevalue of the stopping redundancy of MDS codes along the way,many interesting questions remain unanswered. For example,what is the asymptotic value of for a fixed ?And how does compare with in general?(Do they differ by at most a constant factor?) Finally, is it truethat the stopping redundancy of an MDS codeequals ?

APPENDIX AASYMPTOTICS OF (8) FOR

Rewrite (8) as

(16)

where

Assuming

we show that the st term

prevails in the sum of (16).It suffices to show that each of the other terms is an

fraction of . For , it is easy to verify that



In general, we have

(17)

For , consider two cases. For, we have

In this case, (17) decreases super-exponentially with , and iscertainly . On the other hand, for ,we show that , by showing that

is monotonically increasing for . Indeed,for , we have

APPENDIX BASYMPTOTICS OF THE BOUND (3) FOR

Let , and . The right-hand side of(3) becomes

(18)

Let . Assume that

( being a special case). We show that thefirst term in (18) prevails. Note that

For , we have

For , we have

Noting further that for

we see that the second term in (18) is at most

Hence, at , the upper bound (3) is asymptotic tofor all such that ,

which implies that for , theupper bound (3), when minimized over , is .

Note that the estimate is not always tight.For example, when , we have shown using a dif-ferent analysis that the upper bound (3), when minimized over

, is in fact . However, note that by keepingjust the term in the sum, the upper bound (3) is at least

where . In particular, this shows that theestimate is tight if is . That is, the

bound (3) is when minimized over , for all.

ACKNOWLEDGMENT

The authors wish to thank Tuvi Etzion for helpful discussions.

REFERENCES

[1] Y. Kou, S. Lin, and M. P. C. Fossorier, “Low-density parity-checkcodes based on finite geometries: A rediscovery and new results,” IEEETrans. Inf. Theory, vol. 47, no. 7, pp. 2711–2736, Nov. 2001.

[2] C. Kelley and D. Sridhara, “Pseudocodewords of Tanner graphs,” IEEETrans. Inf. Theory, vol. 53, no. 11, pp. 4013–4038, Nov. 2007.

[3] T. Hehn, J. B. Huber, S. Laendner, and O. Milenkovic, “Multiple-basesbelief-propagation for decoding of short block codes,” in Proc. IEEEInt. Symp. Information Theory, Nice, France, Jun. 2007, pp. 311–315.

[4] C. Di, D. Proletti, I. Telatar, T. Richardson, and R. Urbanke, “Finitelength analysis of low-density parity-check codes,” IEEE Trans. Inf.Theory, vol. 48, no. 6, pp. 1570–1579, Jun. 2002.



[5] M. Schwartz and A. Vardy, “On the stopping distance and stoppingredundancy of codes,” in Proc. IEEE Int. Symp. Information Theory,Adelaide, Australia, Sep. 2005, pp. 975–979.

[6] M. Schwartz and A. Vardy, “On the stopping distance and the stop-ping redundancy of codes,” IEEE Trans. Inf. Theory, vol. 52, no. 3, pp.922–932, Mar. 2006.

[7] T. Etzion, “On the stopping redundancy of Reed–Muller codes,” IEEETrans. Inf. Theory, vol. 52, no. 11, pp. 4867–4879, Nov. 2006.

[8] J. Han and P. H. Siegel, “Improved upper bounds on stopping redun-dancy,” IEEE Trans. Inf. Theory, vol. 53, no. 1, pp. 90–104, Jan. 2007.

[9] O. Milenkovic, E. Soljanin, and P. Whiting, “Stopping and trappingsets in generalized covering arrays,” in Proc. 40th Annu. Conf. Infor-mation Sciences and Systems (CISS), Princeton, NJ, Mar. 2006, pp.259–264.

[10] T. Hehn, S. Laendner, O. Milenkovic, and J. B. Huber, “The stoppingredundancy hierarchy of cyclic codes,” in Proc. 44th Annu. AllertonConf. Communication, Control and Computing, Monticello, IL, Sep.2006, pp. 1271–1280.

[11] K. A. Abdel-Ghaffar and J. H. Weber, “Generalized iterative decodingfor linear block codes on the binary erasure channel,” in Proc. IEEEInt. Symp. Information Theory, Nice, France, Jun. 2007, pp. 66–70.

[12] J. H. Weber and K. A. Abdel-Ghaffar, “Stopping set analysis for Ham-ming codes,” in Proc. IEEE ITSOC Information Theory Workshop onCoding and Complexity, Rotorua, New Zealand, Aug./Sep. 2005, pp.244–247.

[13] J. H. Weber and K. A. S. Abdel-Ghaffar, “Results on parity-check ma-trices with optimal stopping and or dead-end set enumerators,” IEEETrans. Inf. Theory, vol. 54, no. 3, pp. 1368–1374, Mar. 2008.

[14] H. D. L. Hollmann and L. M. G. M. Tolhuizen, “Generic erasure-cor-recting sets: Bounds and constructions,” J. Combin. Theory Ser. A, vol.113, pp. 1746–1759, Nov. 2006.

[15] H. D. L. Hollmann and L. M. G. M. Tolhuizen, “On parity check collec-tions for iterative erasure decoding that correct all correctable erasurepatterns of a given size,” IEEE Trans. Inf. Theory, vol. 53, no. 2, pp.823–828, Feb. 2007.

[16] R. C. Singleton, “Maximum distance �-nary codes,” IEEE Trans. Inf.Theory, vol. IT-10, no. 2, pp. 116–118, Apr. 1964.

[17] I. S. Reed and G. Solomon, “Polynomial codes over certain finitefields,” SIAM J. Appl. Math., vol. 8, no. 2, pp. 300–304, Jun. 1960.

[18] A. Sidorenko, “Upper bounds for Turán numbers,” J. Combin. TheorySer. A, vol. 77, pp. 134–147, 1997.

[19] W. H. Mills and R. C. Mullin, “Coverings and packings,” in Contempo-rary Design Theory, J. H. Dinitz and D. R. Stinson, Eds. New York:Wiley, 1992, ch. 9, pp. 371–399.

[20] J. Han and P. H. Siegel, “On the stopping redundancy of MDS codes,”in Proc. IEEE Int. Symp. Information Theory, Seattle, WA, Jul. 2006,pp. 2491–2495.

[21] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics,2nd ed. Reading, MA: Addison-Wesley, 1994.

[22] G. Katona, T. Nemetz, and M. Simonovits, “On a graph problem ofTurán,” in Mat. Lapok, 1964, vol. 15, pp. 228–238.

[23] J. Han, P. H. Siegel, and A. Vardy, “Improved probabilistic boundson stopping redundancy,” IEEE Trans. Inf. Theory, vol. 54, no. 4, pp.1749–1753, Apr. 2008.

[24] K. H. Kim and F. W. Roush, “On a problem of Turán,” in Studiesin Pure Mathematics: To the Memory of Paul Turán, P. Erdös, Ed.Basel, Switzerland: Birkhäuser-Verlag, 1983, pp. 423–425.

[25] P. Frankl and V. Rödl, “Lower bounds for Turán’s problem,” GraphsCombin., vol. 1, pp. 213–216, 1985.

[26] P. Erdös and J. H. Spencer, Probabilistic Methods in Combinatorics.New York: Academic, 1974.

[27] N. N. Kuzjurin, “Minimal coverings and maximal coverings of �� -subsets by �-subsets,” Matematicheskije Zametki, vol. 21, no. 4, pp.565–571, Apr. 1977.

[28] J. Schönheim, “On coverings,” Pacific J. Math., vol. 14, pp. 1405–1411,1964.

[29] D. de Caen, “Extension of a theorem of Moon and Moser on completesubgraphs,” Ars Combin., vol. 16, pp. 5–10, 1983.

[30] V. Rödl, “On a packing and covering problem,” Europ. J. Combin., vol.5, pp. 69–78, 1985.

[31] W. Mantel, “Vraagstuk XXVIII,” Wiskundige Opgaven met deOplossingen, vol. 10, pp. 60–61, 1907.

[32] P. Turán, “An extremal problem in graph theory,” in Collected Papersof Paul Turán, P. Erdös, Ed. Budapest, Hungary: Akadémiai Kiadó,1990, pp. 231–256.

Junsheng Han (M’08) received the B.Eng. degree in automation from TsinghuaUniversity, Beijing, China, in 1998, the M.S. degree in electrical engineeringfrom the Ohio State University, Columbus, OH, in 2001, and the Ph.D. degreein electrical engineering from University of California, San Diego, La Jolla, CA,in 2008.

He was with Ericsson Wireless Communications, San Diego, CA, from 2001to 2003. Since 2008, he has been with Qualcomm Inc., San Diego, where he iscurrently a Senior Engineer at Corporate R & D. His research interests includecoding theory and communications.

Paul H. Siegel (M’82–SM’90–F’97) received the S.B. and Ph.D. degrees inmathematics from the Massachusetts Institute of Technology (MIT), Cam-bridge, in 1975 and 1979, respectively.

He held a Chaim Weizmann Postdoctoral Fellowship at the Courant Institute,New York University. He was with the IBM Research Division in San Jose, CA,from 1980 to 1995. He joined the faculty of the School of Engineering at the Uni-versity of California, San Diego, in July 1995, where he is currently Professor ofElectrical and Computer Engineering. He is affiliated with the California Insti-tute of Telecommunications and Information Technology, the Center for Wire-less Communications, and the Center for Magnetic Recording Research, wherehe holds an endowed chair and currently serves as Director. His primary researchinterests lie in the areas of information theory and communications, particularlycoding and modulation techniques, with applications to digital data storage andtransmission.

Prof. Siegel was a member of the Board of Governors of the IEEE InformationTheory Society from 1991 to 1996 and was re-elected for a three-year term in2009. He served as Co-Guest Editor of the May 1991 Special Issue on “Codingfor Storage Devices” of the IEEE TRANSACTIONS ON INFORMATION THEORY.He served the same TRANSACTIONS as Associate Editor for Coding Techniquesfrom 1992 to 1995, and as Editor-in-Chief from July 2001 to July 2004. He wasalso Co-Guest Editor of the May/September 2001 two-part issue on “The TurboPrinciple: From Theory to Practice” of the IEEE JOURNAL ON SELECTED AREAS

IN COMMUNICATIONS. He was corecipient, with R. Karabed, of the 1992 IEEEInformation Theory Society Paper Award and shared the 1993 IEEE Communi-cations Society Leonard G. Abraham Prize Paper Award with B. Marcus and J.K. Wolf. With J. B. Soriaga and H. D. Pfister, he received the 2007 Best PaperAward in Signal Processing and Coding for Data Storage from the Data StorageTechnical Committee of the IEEE Communications Society. He holds severalpatents in the area of coding and detection, and was named a Master Inventorat IBM Research in 1994. He is a member of Phi Beta Kappa and the NationalAcademy of Engineering.

Ron M. Roth (M’88–SM’97–F’03) was born in Ramat-Gan, Israel, in 1958.He received the B.Sc. degree in computer engineering, the M.Sc. in electricalengineering, and the D.Sc. degree in computer science from Technion–IsraelInstitute of Technology, Haifa, Israel, in 1980, 1984, and 1988, respectively.

Since 1988, he has been with the Computer Science Department at Technion,where he now holds the General Yaakov Dori Chair in Engineering. Duringthe academic years 1989–1991, he was a Visiting Scientist at IBM ResearchDivision, Almaden Research Center, San Jose, CA, and during 1996–1997 and2004–2005, he was on sabbatical leave at Hewlett-Packard Laboratories, PaloAlto, CA. He is the author of the book Introduction to Coding Theory, publishedby Cambridge University Press in 2006. His research interests include codingtheory, information theory, and their application to the theory of complexity.

Prof. Roth was an Associate Editor for Coding Theory for IEEETRANSACTIONS ON INFORMATION THEORY from 1998 to 2001, and he isnow serving as an Associate Editor for SIAM Journal on Discrete Mathematics.


ieee transactions on information theory, vol. 55, no. 9, september 2009...

Documents