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Coding Schemes for Crisscross Error Patterns Simon Plass, Gerd Richter, and A.J. Han Vinck

Coding Schemes for Crisscross Error Patterns Simon Plass, Gerd Richter, and A.J. Han Vinck. What are Crisscross Errors?.

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Coding Schemes for Crisscross Error Patterns Simon Plass, Gerd Richter, and A.J. Han Vinck

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  1. Coding Schemes for Crisscross Error PatternsSimon Plass, Gerd Richter, and A.J. Han Vinck

  2. 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. Motivation Are there coding scheme which are suited to these crisscross errors? • Rank-Codes • Permutation Codes

  4. Introduction of Rank-Codes Let us consider a vector with elements of the extension field GF(qN): 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. 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. Example of Rank Error 1 = error • Rank array is 2. • rank error = 2 Rank of array is still 2.

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

  8. 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

  9. Key Equation of Rank-Codes Syndrome Sjcan be represented by an appropriate designed shift-register if is known Main problem: Solve the key equation for the unknown variables

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

  11. 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. 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.

  13. 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 0 0 1 0 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1 0

  14. 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. time frequency 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 0 f2 1 0 0 0 f3 0 0 0 1 f4 0 0 1 0 time

  16. 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 10 0 0 1 1 0 0 00 0 0 00 0 1 0 0 0 0 1 impulsive fading narrowband Influence of Different Noise

  17. 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. Thank you!

  19. Error Pattern Example error single error RS codeword

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