List decoding of matrix-product codes from nested codes: An application to quasi-cyclic codes

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August 2012, 6(3): 259-272. doi: 10.3934/amc.2012.6.259

List decoding of matrix-product codes from nested codes: An application to quasi-cyclic codes

Fernando Hernando ^1, , Tom Høholdt ^2, and Diego Ruano ^3,

1.	Department of Mathematics, Universidad Jaume I, Campus Riu Sec, 12071, Castellón de la plana, Spain
2.	DTU-Mathematics, Technical University of Denmark, Matematiktorvet, Building 303, 2800 Kgs. Lyngby, Denmark
3.	Department of Mathematical Sciences, Aalborg University, Fr. Bajersvej 7G, 9920-Aalborg Øst, Denmark

Received March 2011 Revised January 2012 Published August 2012

Abstract
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A list decoding algorithm for matrix-product codes is provided when $C_1, ..., C_s$ are nested linear codes and $A$ is a non-singular by columns matrix. We estimate the probability of getting more than one codeword as output when the constituent codes are Reed-Solomon codes. We extend this list decoding algorithm for matrix-product codes with polynomial units, which are quasi-cyclic codes. Furthermore, it allows us to consider unique decoding for matrix-product codes with polynomial units.

Keywords: list decoding, quasi-cyclic code., Linear code, matrix-product code.

Mathematics Subject Classification: Primary: 94B05; Secondary: 94B3.

Citation: Fernando Hernando, Tom Høholdt, Diego Ruano. List decoding of matrix-product codes from nested codes: An application to quasi-cyclic codes. Advances in Mathematics of Communications, 2012, 6 (3) : 259-272. doi: 10.3934/amc.2012.6.259

References:

[1]	P. Beelen and K. Brander, Key equations for list decoding of Reed-Solomon codes and how to solve them,, J. Symbolic Comput., 45 (2010), 773. doi: 10.1016/j.jsc.2010.03.010. Google Scholar
[2]	T. Blackmore and G. H. Norton, Matrix-product codes over $\mathbb F_q$,, Appl. Algebra Engrg. Comm. Comput., 12 (2001), 477. doi: 10.1007/PL00004226. Google Scholar
[3]	I. I. Dumer, Concatenated codes and their multilevel generalizations,, in, (1998), 1911. Google Scholar
[4]	P. Elias, List decoding for noisy channels,, Rep. No. 335, (1957). Google Scholar
[5]	V. Guruswami and A. Rudra, Better binary list decodable codes via multilevel concatenation,, IEEE Trans. Inform. Theory, 55 (2009), 19. doi: 10.1109/TIT.2008.2008124. Google Scholar
[6]	V. Guruswami and M. Sudan, Improved decoding of Reed-Solomon and algebraic-geometry codes,, IEEE Trans. Inform. Theory, 45 (1999), 1757. doi: 10.1109/18.782097. Google Scholar
[7]	F. Hernando, K. Lally and D. Ruano, Construction and decoding of matrix-product codes from nested codes,, Appl. Algebra Engrg. Comm. Comput., 20 (2009), 497. doi: 10.1007/s00200-009-0113-5. Google Scholar
[8]	F. Hernando and D. Ruano, New linear codes from matrix-product codes with polynomial units,, Adv. Math. Commun., 4 (2010), 363. doi: 10.3934/amc.2010.4.363. Google Scholar
[9]	T. Kasami, A Gilbert-Varshamov bound for quasi-cyclic codes of rate 1/2,, IEEE Trans. Inform. Theory, IT-20 (1974). doi: 10.1109/TIT.1974.1055262. Google Scholar
[10]	K. Lally, Quasicyclic codes - some practical issues,, in, (2002). Google Scholar
[11]	K. Lally and P. Fitzpatrick, Algebraic structure of quasicyclic codes,, Discrete Appl. Math., 111 (2001), 157. doi: 10.1016/S0166-218X(00)00350-4. Google Scholar
[12]	K. Lee and M. E. O'Sullivan, List decoding of Reed-Solomon codes from a Gröbner basis perspective,, J. Symbolic Comput., 43 (2008), 645. doi: 10.1016/j.jsc.2008.01.002. Google Scholar
[13]	R. R. Nielsen and T. Høholdt, Decoding Reed-Solomon codes beyond half the minimum distance,, in, (2000), 221. Google Scholar
[14]	F. Özbudak and H. Stichtenoth, Note on Niederreiter-Xing's propagation rule for linear codes,, Appl. Algebra Engrg. Comm. Comput., 13 (2002), 53. doi: 10.1007/s002000100091. Google Scholar
[15]	W. C. Schmid and R. Schürer, "Mint,'', Dept. of Mathematics, (). Google Scholar
[16]	J. M. Wozencraft, List decoding,, in, (1958), 90. Google Scholar

show all references

References:

[1]	P. Beelen and K. Brander, Key equations for list decoding of Reed-Solomon codes and how to solve them,, J. Symbolic Comput., 45 (2010), 773. doi: 10.1016/j.jsc.2010.03.010. Google Scholar
[2]	T. Blackmore and G. H. Norton, Matrix-product codes over $\mathbb F_q$,, Appl. Algebra Engrg. Comm. Comput., 12 (2001), 477. doi: 10.1007/PL00004226. Google Scholar
[3]	I. I. Dumer, Concatenated codes and their multilevel generalizations,, in, (1998), 1911. Google Scholar
[4]	P. Elias, List decoding for noisy channels,, Rep. No. 335, (1957). Google Scholar
[5]	V. Guruswami and A. Rudra, Better binary list decodable codes via multilevel concatenation,, IEEE Trans. Inform. Theory, 55 (2009), 19. doi: 10.1109/TIT.2008.2008124. Google Scholar
[6]	V. Guruswami and M. Sudan, Improved decoding of Reed-Solomon and algebraic-geometry codes,, IEEE Trans. Inform. Theory, 45 (1999), 1757. doi: 10.1109/18.782097. Google Scholar
[7]	F. Hernando, K. Lally and D. Ruano, Construction and decoding of matrix-product codes from nested codes,, Appl. Algebra Engrg. Comm. Comput., 20 (2009), 497. doi: 10.1007/s00200-009-0113-5. Google Scholar
[8]	F. Hernando and D. Ruano, New linear codes from matrix-product codes with polynomial units,, Adv. Math. Commun., 4 (2010), 363. doi: 10.3934/amc.2010.4.363. Google Scholar
[9]	T. Kasami, A Gilbert-Varshamov bound for quasi-cyclic codes of rate 1/2,, IEEE Trans. Inform. Theory, IT-20 (1974). doi: 10.1109/TIT.1974.1055262. Google Scholar
[10]	K. Lally, Quasicyclic codes - some practical issues,, in, (2002). Google Scholar
[11]	K. Lally and P. Fitzpatrick, Algebraic structure of quasicyclic codes,, Discrete Appl. Math., 111 (2001), 157. doi: 10.1016/S0166-218X(00)00350-4. Google Scholar
[12]	K. Lee and M. E. O'Sullivan, List decoding of Reed-Solomon codes from a Gröbner basis perspective,, J. Symbolic Comput., 43 (2008), 645. doi: 10.1016/j.jsc.2008.01.002. Google Scholar
[13]	R. R. Nielsen and T. Høholdt, Decoding Reed-Solomon codes beyond half the minimum distance,, in, (2000), 221. Google Scholar
[14]	F. Özbudak and H. Stichtenoth, Note on Niederreiter-Xing's propagation rule for linear codes,, Appl. Algebra Engrg. Comm. Comput., 13 (2002), 53. doi: 10.1007/s002000100091. Google Scholar
[15]	W. C. Schmid and R. Schürer, "Mint,'', Dept. of Mathematics, (). Google Scholar
[16]	J. M. Wozencraft, List decoding,, in, (1958), 90. Google Scholar

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