American Institute of Mathematical Sciences

Advanced
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

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

[1]

Irene Márquez-Corbella, Edgar Martínez-Moro, Emilio Suárez-Canedo. On the ideal associated to a linear code. Advances in Mathematics of Communications, 2016, 10 (2) : 229-254. doi: 10.3934/amc.2016003
[2]

Laura Luzzi, Ghaya Rekaya-Ben Othman, Jean-Claude Belfiore. Algebraic reduction for the Golden Code. Advances in Mathematics of Communications, 2012, 6 (1) : 1-26. doi: 10.3934/amc.2012.6.1
[3]

Serhii Dyshko. On extendability of additive code isometries. Advances in Mathematics of Communications, 2016, 10 (1) : 45-52. doi: 10.3934/amc.2016.10.45
[4]

Fernando Hernando, Diego Ruano. New linear codes from matrix-product codes with polynomial units. Advances in Mathematics of Communications, 2010, 4 (3) : 363-367. doi: 10.3934/amc.2010.4.363
[5]

Olof Heden. The partial order of perfect codes associated to a perfect code. Advances in Mathematics of Communications, 2007, 1 (4) : 399-412. doi: 10.3934/amc.2007.1.399
[6]

Selim Esedoḡlu, Fadil Santosa. Error estimates for a bar code reconstruction method. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1889-1902. doi: 10.3934/dcdsb.2012.17.1889
[7]

Sascha Kurz. The $[46, 9, 20]_2$ code is unique. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020074
[8]

Michael Kiermaier, Johannes Zwanzger. A $\mathbb Z$4-linear code of high minimum Lee distance derived from a hyperoval. Advances in Mathematics of Communications, 2011, 5 (2) : 275-286. doi: 10.3934/amc.2011.5.275
[9]

M. Delgado Pineda, E. A. Galperin, P. Jiménez Guerra. MAPLE code of the cubic algorithm for multiobjective optimization with box constraints. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 407-424. doi: 10.3934/naco.2013.3.407
[10]

Andrew Klapper, Andrew Mertz. The two covering radius of the two error correcting BCH code. Advances in Mathematics of Communications, 2009, 3 (1) : 83-95. doi: 10.3934/amc.2009.3.83
[11]

Masaaki Harada, Takuji Nishimura. An extremal singly even self-dual code of length 88. Advances in Mathematics of Communications, 2007, 1 (2) : 261-267. doi: 10.3934/amc.2007.1.261
[12]

José Gómez-Torrecillas, F. J. Lobillo, Gabriel Navarro. Information--bit error rate and false positives in an MDS code. Advances in Mathematics of Communications, 2015, 9 (2) : 149-168. doi: 10.3934/amc.2015.9.149
[13]

Jorge P. Arpasi. On the non-Abelian group code capacity of memoryless channels. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020058
[14]

M. De Boeck, P. Vandendriessche. On the dual code of points and generators on the Hermitian variety $\mathcal{H}(2n+1,q^{2})$. Advances in Mathematics of Communications, 2014, 8 (3) : 281-296. doi: 10.3934/amc.2014.8.281
[15]

Sihuang Hu, Gabriele Nebe. There is no $[24,12,9]$ doubly-even self-dual code over $\mathbb F_4$. Advances in Mathematics of Communications, 2016, 10 (3) : 583-588. doi: 10.3934/amc.2016027
[16]

Anna-Lena Horlemann-Trautmann, Kyle Marshall. New criteria for MRD and Gabidulin codes and some Rank-Metric code constructions. Advances in Mathematics of Communications, 2017, 11 (3) : 533-548. doi: 10.3934/amc.2017042
[17]

Masaaki Harada, Ethan Novak, Vladimir D. Tonchev. The weight distribution of the self-dual $[128,64]$ polarity design code. Advances in Mathematics of Communications, 2016, 10 (3) : 643-648. doi: 10.3934/amc.2016032
[18]

Denis S. Krotov, Patric R. J.  Östergård, Olli Pottonen. Non-existence of a ternary constant weight $(16,5,15;2048)$ diameter perfect code. Advances in Mathematics of Communications, 2016, 10 (2) : 393-399. doi: 10.3934/amc.2016013
[19]

Jianying Fang. 5-SEEDs from the lifted Golay code of length 24 over Z4. Advances in Mathematics of Communications, 2017, 11 (1) : 259-266. doi: 10.3934/amc.2017017
[20]

Pedro Branco. A post-quantum UC-commitment scheme in the global random oracle model from code-based assumptions. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020046

2018 Impact Factor: 0.879

Tools

Metrics

  • PDF downloads (41)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences