8/14/2019 A Novel Approach to Improving Burst Errors Correction Capability of Hamming Code

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A Novel Approach to Improving

Burst Errors Correction

Capability of Hamming Code

: : 2009.12.15

Communications, Circuits and Systems, 2007. ICCCAS2007. International Conference on

Jianwu Zhao Yibing Shi

Univ. of Electron. Sci. & Technol. of China, Chengdu

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Abstract & Introduction

Extended Hamming Code

Design AlgorithmsExample

Conclusion

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Abstract & Introduction

Error detection and correction is critical to accurate

data transmission, storage and retrieval.

In high-reliability applications, the single error

correction and double error detections(SEC-DED)

Hamming code may not provide adequate protection

against burst errors

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Abstract & Introduction

This makes multiple-error correction (MEC) highly

desirable

In order to detect or correct transmission or storageerrors, some additional, redundant bits, called "check

bits", can be added to a number of data bits to construct

a codeword

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Abstract & Introduction

Our concern in this paper is exclusively

binary forward error correcting (FEC) block

codes in data transmission

Example : (7,4,3) Hamming code

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Abstract & Introduction

This paper proposed a novel approach to

improving burst errors correction capability of

the extended Hamming Code , with the target

objective of minimizing the redundancy, while

retaining code rate as maximum as possible

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Extended Hamming Code

Hamming codeword

Hamming ruler bits can distinguish cases

k : the number of data bit

r : the number of additional parity check bits

n : the size of codeword , n = k+r

Notation : ( n , k , r )

r2

12 ++ rkr

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Extended Hamming Code

=

101010101010101

110011001100110

111100001111000

111111110000000

H

Original form of matrix H:

(16, 11, 4) extended Hamming code

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Extended Hamming Code

=

1000011011010101

0100010110110011

0010001110001111

0001000001111111

1111111111111111

G

Identity-check matrix of the(16, 11, 4) extended

Hamming code

The C0=0xAB61, C1=0xCDA2, C2=0xFlC4, C3=0xFE08,

and C4=0xFFFF.

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Extended Hamming Code

For any integer p there exists a (n, k, r)

extended Hamming code, where:

12 = pn

pk p = 12

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Extended Hamming Code

Step 1) Create a matrix with dimension ,

which each column is to be different, i.e.

all binary combinations should be used,

except all-zero's one

Step 2) Reorder the columns in order to achieve

an identity matrix in the right hand side

( )12 pp

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Extended Hamming Code

Step 3) Take first columns of the

matrix (all columns except those

creating the identity part), and then

add a zero column (overall parity-

check column) to this matrix on the

left.

Step 4) Append this matrix with an overallparity-check column to an identity

matrix with dimensions

( ) pp 12

pp

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Extended Hamming Code

Step 5) Add a row of all l's to the matrix with

dimensions on the top

Step 6) The check code is equal to the value

of each row of the extended identity

check matrix

( ) pp 21 +

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Design Algorithms

For data transmission on serial channels, the

most prevalent errors are multi-bit bursts.

The proposed approach to correct burst error

is the interleaving reorganization of Hamming

codewords

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Design Algorithms

These advantages, however, come at the

cost of failing to correct multiple single-bit

errors in separate byte, which result in

consecutive bits error in a code word

This technique degrades the ability to correct

single bit error (random errors) in Hammingcodewords

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Algorithm for the encoding

Step 1) Take out the 11 bytes data from the

original data and then get 8 data

with 11 bit width by simply merging.

Step 2) Compute the values of even check bits

and overall even check bit from the

check codes and then respectively put

these even check bits in position 1, 2,4,8, and 16.

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Algorithm for the encoding

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1

0 0 0 0 0 0 1 0

0 0 0 0 0 0 1 1

0 0 0 0 0 1 0 0

0 0 0 0 0 1 0 1

0 0 0 0 0 1 1 0

0 0 0 0 0 1 1 1

0 0 0 0 1 0 0 0

0 0 0 0 1 0 0 1

0 0 0 0 1 0 1 0

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Algorithm for the encoding

Step 3) Repeat Step 2. Process other 11 bits

data until all data are encoded.

Step 4) Reorganize the Hamming codewords

by the use of interleaving technique

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Example

Data Class Physical Memory Area

Original Testing Data 0x2000 0x200A

Encoded Data 0x2020 0x202F

Interleaved Data 0x2040 0x204F

Decoded Data 0x2060 0x206A

Memory Assignments for Testing Program

Hamming encoding and decoding testing program in assembly

language and assume the use of 8-bit microcomputer AT89C55.

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Example

Without using interleaving reorganization technique

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Example

Using interleaving reorganization technique

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Example

Using interleaving reorganization technique

suppose that the data 0x0F is sent and burst errors occur such

that the data 0xF0 is received.

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Conclusion

In the paper, we analyze the advantage and

disadvantage of the Hamming Codes and

error mechanisms causing random errors and

burst errors.

Because burst errors can be effectively

randomized by the use of interleaving, burst

errors correction capability of Hammingcodes is greatly improved.