Coding Schemes for Crisscross Error Patterns
Simon Plass, Gerd Richter, and A.J. Han Vinck
Gothenburg, 11-12 April, 2007
German
Aerospace Center
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):
x  ( x1 , x2 , , xn )
Now, we can present the vector x as a matrix with entries of the finite field GF(q):
 a1,1 a1,2

a2,1 a2,2

A( x) 


 aN ,1 aN ,1
a1, n 

a2, n 


aN ,n 
Let us define the rank distance between two matrices A and B as:
dr ( A, B)  rank ( A  B)
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Introduction of Rank-Codes (cont’d)
Example for the rank distance:
1
1
d r ( A, B)  rank ( 
0

0
1 1 1  0
1 0 0  1

0 0 0  0
 
1 1 0  0
0 0 0
1
0
0 0 0 
)  rank ( 
0
1 0 0


0 1 0
0
1 1 1
1 0 0 
)2
1 0 0

1 0 0
Furthermore, Rank-Codes have an error correction capability t of
 d  1
rank ( E )  t   r 
 2 
where E is the error matrix.
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Example of Rank Error
1 = error
Rank array is 2.
rank error = 2
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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:
 h1
 hq
 1
2
H   h1q


 q d 2
 h1
h2
h3
h2q
h3q
h2q
2
q d 2
2
h
h3q
2
q d 2
3
h
The elements h1 , h2 ,
hn 
hnq 
2
hnq 


q d 2 
hn 
, hn  GF (q N )
 g1
 gq
 1
2
G   g1q


 qk 1
 g1
and g1 , g 2 ,
g2
g3
g 2q
g3q
g 2q
g 2q
2
k 1
g3q
g3q
2
k 1
gn 
g nq 
2
g nq 


k 1 
g nq 
, g n  GF (q N )
must be linearly independent over GF (q N ).
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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 computation
by recursive calculation
 Error vector e
cdecode = r - e
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Key Equation of Rank-Codes
 S0q
 
 S1q
 q
 S2


 S q
  1
 
     S 

q1
S    1    S 1 

q1
S 1     2     S  2 

 


 

1    S 2 1 
q1  

S2  2 

q1
 1
S

S j    i S ,
i 1
qi
j i
Syndrome Sj can
be represented by an
appropriate designed
shift-register if 
i
is known
j   , , 2  1
Main problem: Solve the key equation for the unknown variables  i .
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Berlekamp-Massey Algorithm for Rank-Codes
Initialize the algorithm
New theorem and proof
Does current design of
shift-register produce next
syndrome?
No
Yes
Modify shift-register
Has shift-register correct length?
Yes
No
Modify length
No
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All syndromes calculated?
Yes and finished
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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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Introduction of Permutation Codes
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.
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Example of a simple Permutation Code
N=3, dmin=2, |C|=6 and the resulting codewords:
123
231
312
213
321
132
001
100
010
010
001
100
010
100
001
001
010
100
100
001
010
As binary matrix:
100
010
001
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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};
time
frequency
{2143}  {f2 f1 f4 f3} 
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f1
f2
f3
f4
0100
1000
0001
0010
time
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Influence of Different Noise
1000
0100
0010
0001
No noise
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1010
0100
0010
0001
1000
0000
0010
0001
Background noise
1111
0100
0010
0001
1001
0101
0011
0001
1000
0000
0010
0001
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
error
single error
RS codeword
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