1. Ger man Aerospace Center Gothenburg, 11-12 April, 2007 Coding Schemes for Crisscross Error Patterns Simon Plass, Gerd Richter, and A.J. Han Vinck

2. 2 Ger man 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.

3. 3 Ger man Aerospace Center Motivation Are there coding scheme which are suited to these crisscross errors?  Rank-Codes  Permutation Codes

4. 4 Ger man Aerospace Center Introduction of Rank-Codes Let us consider a vector with elements of the extension field GF(q N ): Now, we can present the vector x as a matrix with entries of the finite field GF(q): Let us define the rank distance between two matrices A and B as:

5. 5 Ger man Aerospace Center Introduction of Rank-Codes (cont’d) Example for the rank distance: Furthermore, Rank-Codes have an error correction capability t of where E is the error matrix.

6. 6 Ger man Aerospace Center Example of Rank Error 1 = error Rank array is 2.  rank error = 2 Rank of array is still 2.

7. 7 Ger man Aerospace Center Construction of Rank-Codes A parity-check matrix H and its corresponding generator matrix G which define the Rank-Code are given by: The elementsand must be linearly independent over

8. 8 Ger man Aerospace Center Algebraic Decoding Syndrome calculation s=(c+e)H T =eH T  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 c decode = r - e

9. 9 Ger man Aerospace Center Key Equation of Rank-Codes Main problem: Solve the key equation for the unknown variables Syndrome S j can be represented by an appropriate designed shift-register if is known

10. 10 Ger man Aerospace Center Berlekamp-Massey Algorithm for Rank-Codes Initialize the algorithm Does current design of shift-register produce next syndrome? Modify shift-register Has shift-register correct length? Modify length All syndromes calculated? Yes No Yes No Yes and finished New theorem and proof

11. 11 Ger man Aerospace Center 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

12. 12 Ger man Aerospace Center 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

13. 13 Ger man Aerospace Center Example of a simple Permutation Code N=3, d min =2, |C|=6 and the resulting codewords: 1 2 32 3 13 1 22 1 33 2 11 3 2 As binary matrix: 1 0 00 0 10 1 00 1 00 0 11 0 0 0 1 01 0 00 0 11 0 00 1 00 0 1 0 0 10 1 01 0 00 0 11 0 00 1 0

14. 14 Ger man Aerospace Center 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

15. 15 Ger man Aerospace Center 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}  {f 2 f 1 f 4 f 3 }  f 1 0 1 0 0 f 2 1 0 0 0 f 3 0 0 0 1 f 4 0 0 1 0 time frequency time

16. 16 Ger man Aerospace Center Influence of Different Noise 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 No noise Background noise 1 1 1 1 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 1 0 1 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 narrowband impulsivefading

17. 17 Ger man Aerospace Center 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

18. 18 Ger man Aerospace Center Thank you!

19. 19 Ger man Aerospace Center Error Pattern Example RS codeword single error error