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Gothenburg, 11-12 April, 2007

Coding Schemes for Crisscross Error Patterns

Simon Plass, Gerd Richter, and A.J. Han Vinck

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What are Crisscross Errors?

Crisscross errors can occur in several applications of information transmission, e.g., magnetic tape recording, memory chip arrays or in environments with impulsive- or narrowband noise, where the information is stored or transmitted in (N x n) arrays.

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Motivation

Are there coding scheme which are suited to these crisscross errors?

Rank-Codes

Permutation Codes

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Introduction of Rank-Codes

Let us consider a vector with elements of the extension field GF(qN):

1 2( , , , )nx x x x Now, we can present the vector x as a matrix with entries of the finite field GF(q):

1,1 1,2 1,

2,1 2,2 2,

,1 ,1 ,

( )

n

n

N N N n

a a aa a a

A x

a a a

Let us define the rank distance between two matrices A and B as:

( , ) ( )rd A B rank A B

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Introduction of Rank-Codes (cont’d)

Example for the rank distance:

1 1 1 1 0 0 0 0 1 1 1 11 1 0 0 1 0 0 0 0 1 0 0

( , ) ( ) ( ) 20 0 0 0 0 1 0 0 0 1 0 00 1 1 0 0 0 1 0 0 1 0 0

rd A B rank rank

Furthermore, Rank-Codes have an error correction capability t of

where E is the error matrix.

1( )2rdrank E t

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Example of Rank Error

1 = error

Rank array is 2.rank error = 2

Rank of array is still 2.

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Construction of Rank-Codes

A parity-check matrix H and its corresponding generator matrix G which define the Rank-Code are given by:

2 2 2 2

2 2 2 2

1 2 3

1 2 3

1 2 3

1 2 3

d d d d

nq q q q

n

q q q qn

q q q qn

h h h hh h h h

H h h h h

h h h h

2 2 2 2

1 1 1 1

1 2 3

1 2 3

1 2 3

1 2 3

k k k k

nq q q q

n

q q q qn

q q q qn

g g g gg g g g

G g g g g

g g g g

The elements 1 2, , , ( )Nnh h h GF q and 1 2, , , ( )Nng g g GF q

must be linearly independent over ( ).NGF q

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Algebraic Decoding

Syndrome calculation s=(c+e)HT=eHT

Key equation

Use of efficient algorithm,e.g., Berlekamp-Massey algorithm,

for solving the system of linear equations Error polynomial

Error value and error location computationby recursive calculation

Error vector e

cdecode = r - e

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Key Equation of Rank-Codes1

1

1

1

0 1

1 11

222 1

2 111 2 2

q q

q q

q q

q q

S S SS S S

SS S

SS S

Main problem: Solve the key equation for the unknown variables .i1

,iq

j i j ii

S S

, , 2 1j

Syndrome Sj canbe represented by anappropriate designedshift-register if is known

i

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Berlekamp-Massey Algorithm for Rank-Codes

Initialize the algorithm

Does current design ofshift-register produce next

syndrome?

Modify shift-register

Has shift-register correct length?

Modify length

All syndromes calculated?

Yes

No

No

Yes

NoYes and finished

New theorem and proof

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Conclusions for Rank-Codes

Rank-Codes exploit the rank metric by decoding over the rank of the error matrix, and therefore, Rank-Codes can handle efficiently crisscross errors

The Berlekamp-Massey algorithm was introduced as an efficient decoding algorithm

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A Permutation Code C consists of |C| codewords of length N, where every codeword contains the N different integers 1,2,…,N as symbols.

The cardinality |C| is upper bounded by

The codewords are presented in a binary matrix where every row and column contains exactly one single symbol 1.

Introduction of Permutation Codes

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Example of a simple Permutation Code

N=3, dmin=2, |C|=6 and the resulting codewords:

1 2 3 2 3 1 3 1 2 2 1 3 3 2 1 1 3 2

As binary matrix:

1 0 0 0 0 1 0 1 0 0 1 0 0 0 1 1 0 00 1 0 1 0 0 0 0 1 1 0 0 0 1 0 0 0 10 0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1 0

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Influence of Crisscross and Random Errors

A row or column error reduces the distance between two codewords by a maximum value of two.

A random error reduces the distance by a maximum value of one.

We can correct these errors, if

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Application to M-FSK Modulation

In M-FSK, symbols are modulated as one of M orthogonal sinusoidal waves

The setting of Permutation Codes can be mapped onto M-FSK modulation

Example: M=N=4, |C|=4, C={1234}, {2143}, {3412}, {4321};

{2143} {f2 f1 f4 f3} f1 0 1 0 0f2 1 0 0 0f3 0 0 0 1f4 0 0 1 0time

frequency time

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Influence of Different Noise

1 0 0 00 1 0 00 0 1 0 0 0 0 1

1 0 1 00 1 0 00 0 1 0 0 0 0 1

1 0 0 00 0 0 00 0 1 0 0 0 0 1

No noise Background noise

1 1 1 10 1 0 00 0 1 0 0 0 0 1

1 0 0 10 1 0 10 0 1 1 0 0 0 1

1 0 0 00 0 0 00 0 1 0 0 0 0 1

narrowband impulsive fading

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Conclusions

Introduction of codes, namely Rank-Codes and Permutation Codes, which can handle crisscross errors

Rank-Codes:• Rank-Codes exploit the rank metric by decoding over the rank of

the error matrix, and therefore, Rank-Codes can handle efficiently crisscross errors

• The Berlekamp-Massey algorithm was introduced as an efficient decoding algorithm

Permutation Codes:• Binary code for the crisscross error problem• Example of M-FSK modulation application is introduced

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Thank you!

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Error Pattern Example

RS codeword

single error

error