On the decoding of binary cyclic codes with the Newton identities

Journal of Symbolic Computation

Volumn 44, Issue 12, 2009, Pages 1608-1625

  Augot, Daniel ^a   Bardet, Magali ^b   Faugère, Jean Charles ^c  

^a INRIA ROCQUENCOURT   (France)

^b UNIVERSITÉ DE ROUEN   (France)

^c INRIA   (France)

Author keywords

Cyclic codes; Elimination theory; F 4 algorithm; Gr bner bases; Quadratic residue codes

Indexed keywords

EID: 69949135762     PISSN: 07477171     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jsc.2008.02.006     Document Type: Article

References (44)

1. Augot, D., Bardet, M., Faugère, J.-C., 2003. Efficient decoding of (binary) cyclic codes above the correction capacity of the code using Gröbner bases, In: Proceedings of the 2003 IEEE International Symposium on Information Theory, pp. 362-362
2. Augot D., Charpin P., and Sendrier N. Studying the locator polynomial of minimum weight codewords of BCH codes. IEEE Transactions on Information Theory 38 3 (1992) 960-973
3. Bardet, M., 2004. Étude des systémes algébriques surdeterminés. Applications aux codes correcteurs et à la cryptographie. Ph.D. Thesis, Université Paris 6
4. Bardet M., Faugère J.C., and Salvy B. On the complexity of Gröbner basis computation of semi-regular overdetermined algebraic equations. In: Valibouze A. (Ed). ICPSS Paris (2004) 71-75
5. Bardet, M., Faugère, J.C., Salvy, B., 2005. Asymptotic expansion of the degree of regularity for semi-regular systems of equations, In: P. Gianni (Ed.). Mega 2005 Sardinia (Italy), 2005, 15 pages
6. Barg A. Complexity issues in coding theory. In: Pless V.S., and Huffman W.C. (Eds). Handbook of Coding Theory (1998), North-Holland 649-754
7. Buchberger, B., 1965. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, Ph.D. Thesis, Innsbruck
8. Buchberger B. An algorithmical criterion for the solvability of algebraic systems. Aequationes Mathematicae 4 3 (1970) 374-383 (in German)
9. Buchberger B. A theoretical basis for the reduction of polynomials to canonical forms. ACM SIGSAM Bulletin 39 (1976) 19-29
10. Buchberger B. A criterion for detecting unnecessary reductions in the construction of Gröbner basis. Proc. EUROSAM 79. Lect. Notes in Comp. Sci. vol. 72 (1979), Springer-Verlag 3-21
11. Buchberger B. Gröbner Bases : An algorithmic method in polynomial ideal theory. In: Reidel Publishing Company (Ed). Recent trends in multidimensional system theory (1985), Bose 184-232 (chapter 6)
12. Caboara M., and Mora T. The Chen-Reed-Helleseth-Truong decoding algorithm and the Gianni-Kalkbrenner Gröbner shape theorem. Applicable Algebra in Engineering, Communication and Computing 13 3 (2002) 209-232
13. Canteaut A., and Chabaud F. A new algorithm for finding minimum-weight words in a linear code: Application to McEliece's cryptosystem and to narrow-sense BCH codes of length 511. IEEE Transactions on Information Theory 44 1 (1998) 367-378
14. Chang Y., Truong T.-K., Reed I.S., Cheng H.Y., and Lee C.D. Algebraic decoding of (71, 36, 11), (79, 40, 15), and (97, 49, 15) quadratic residue codes. IEEE Transactions on Communications 51 9 (2003) 1463-1473
15. Chen X., Reed I.S., Helleseth T., and Truong T.K. Use of Gröbner bases to decode binary cyclic codes up to the true minimum distance. IEEE Transactions on Information Theory 40 5 (1994) 1654-1661
16. Chen X., Reed I.S., Helleseth T., and Truong T.K. General principles for the algebraic decoding of cyclic codes. IEEE Transactions on Information Theory 40 5 (1994) 1661-1663
17. Chen X., Reed I.S., and Truong T.-K. Decoding the (73, 37, 13) quadratic residue code. IEE Proceedings on Computers and Digital Techniques 141 5 (1994) 253-258
18. Chien R.T. Cyclic decoding procedures for Bose-Chaudhuri-Hocquenghem codes. IEEE Transactions on Information Theory 10 4 (1964) 357-363
19. Cooper III A.B. Direct solution of BCH syndrome equations. In: Arkian E. (Ed). Communications, Control, and Signal Processing (1990), Elsevier 281-286
20. Cooper III A.B. Finding BCH error locator polynomials in one step. Electronics Letters 27 22 (1991) 2090-2091
21. Cooper III, A.B., 1991b. A one-step algorithm for finding BCH error locator polynomials, International Symposium ion Information Theory, ISIT 1991, pp. 93-93
22. Cox D., Little J., and O'Shea D. Ideals, varieties and algorithms (1992), Springer
23. Cox D.A., Little J., and O'Shea D. Using algebraic geometry. Graduate Texts in Mathematics (2005), Springer
24. deBoer M., and Pellikaan R. Gröbner bases for codes. In: Cohen A.M., Cuypers H., and Sterk H. (Eds). Some Tapas in Computer Algebra. Algorithms and Computation in Mathematics vol.4 (1999), Springer 237-259
25. deBoer M., and Pellikaan R. Gröbner bases for decoding. In: Cohen A.M., Cuypers H., and Sterk H. (Eds). Some Tapas in Computer Algebra. Algorithms and Computation in Mathematics vol. 4 (1999), Springer 260-275
26. Eisenbud D. Commutative algebra with a view toward algebraic geometry. Graduate Texts in Mathematics vol. 150 (1995), Springer-Verlag
27. Faugère J.C. A new efficient algorithm for computing Gröbner bases (F4). Journal of Pure and Applied Algebra 139 1-3 (1999) 61-88
28. Faugère J.C. A new efficient algorithm for computing Gröbner bases without reduction to zero F5. In: Mora T. (Ed). Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation (2002), ACM Press. 1-58113-484-3 75-83
29. Gianni P. Properties of Gröbner bases under specializations. In: Davenport J. (Ed). Eurocal '87: European Conference on Computer Algebra, Leipzig GDR. Lecture Notes in Computer Science vol.378 (1989), Springer 293-297
30. Grassl, M., 2000. On the minimum distance of some quadratic-residue codes, 2000 IEEE International Symposium on Information Theory (ISIT), Sorento, pp. 253-253
31. He R., Reed I.S., Truong T.-K., and Chen X. Decoding the (47, 24, 11) quadratic residue code. IEEE Transactions on Information Theory 47 3 (2001) 1181-1186
32. Kalkbrener M. Solving systems of algebraic equations by using Gröbner bases. EUROCAL '87 (Leipzig, 1987). Lecture Notes in Computer Science vol. 378 (1989), Springer 282-292
33. Lidl R., and Niederreiter H. Finite fields. Encyclopedia of Mathematics and its Applications (1996), Cambridge University Press
34. Loustaunau P., and York E.V. On the decoding of cyclic codes using Gröbner bases. Applicable Algebra in Engineering, Communication and Computation 8 6 (1997) 469-483
35. Lu E.H., Wuu H.P., Cheng Y.C., and Lu P.C. Fast algorithms for decoding the (23,12) binary Golay code with four-error-correcting capability. International Journal of Systems Science 26 4 (1995) 937-945
36. Macwilliams F.J., and Sloane N.J.A. The theory of error-correcting codes. North-Holland Mathematical Library (1983), North Holland
37. Orsini E., and Sala M. Correcting errors and erasures via the syndrome variety. Journal of Pure and Applied Algebra 200 1-2 (2005) 191-226
38. Orsini E., and Sala M. General error locator polynomials for binary cyclic codes with t ≤ 2 and n < 63. IEEE Transactions on Information Theory 53 3 (2007) 1095-1107
39. Reed I.S., Truong T.-K., Chen X., and Yin X. The algebraic decoding of the (41, 21, 9) quadratic residue code. IEEE Transactions on Information Theory 38 3 (1992) 974-986
40. Reed I.S., Yin X., and Truong T.-K. Algebraic decoding of the (32, 16, 8) quadratic residue code. IEEE Transactions on Information Theory 36 4 (1990) 876-880
41. Reed I.S., Yin X., Truong T.-K., and Holmes J.K. Decoding the (24,12,8) Golay code. Computers and Digital Techniques, IEE Proceedings- 137 3 (1990) 202-206
42. Tjhai C., Tomlinson M., Grassl M., Horan R., Ahmed M., and Ambroze M. New linear codes derived from binary cyclic codes of length 151. IEE Proceedings on Communications 153 5 (2006) 581-585
43. Truong T.-K., Chang Y., Chen Y.-H., and Lee C.D. Algebraic decoding of (103, 52, 19) and (113, 57, 15) quadratic residue codes. IEEE Transactions on Communications 53 5 (2005) 749-754
44. Valibouze, A., 1995. Modules de Cauchy, polynômes caracteristiques et resolvantes. Tech. Rep. 1995-131, LITP