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Coding Schemes for Crisscross Error Patterns Simon Plass, Gerd Richter, and A.J. Han Vinck. What are Crisscross Errors?. - PowerPoint PPT Presentation
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Ger manAerospace Center
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