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

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.

[1]  K. Lally,et al.  Quasicyclic codes - some practical issues , 2002, Proceedings IEEE International Symposium on Information Theory,.

[2]  Peter Beelen,et al.  Key equations for list decoding of Reed-Solomon codes and how to solve them , 2010, J. Symb. Comput..

[3]  Venkatesan Guruswami,et al.  Improved decoding of Reed-Solomon and algebraic-geometry codes , 1999, IEEE Trans. Inf. Theory.

[4]  Kwankyu Lee,et al.  List decoding of Reed-Solomon codes from a Gröbner basis perspective , 2008, J. Symb. Comput..

[5]  Tadao Kasami.  A Gilbert-Varshamov bound for quasi-cycle codes of rate 1/2 (Corresp.) , 1974, IEEE Trans. Inf. Theory.

[6]  Fernando Hernando,et al.  Construction and decoding of matrix-product codes from nested codes , 2009, Applicable Algebra in Engineering, Communication and Computing.

[7]  Tom Høholdt,et al.  Decoding Reed-Solomon Codes Beyond Half the Minimum Distance , 2000 .

[8]  Fernando Hernando,et al.  New linear codes from matrix-product codes with polynomial units , 2010, Adv. Math. Commun..

[9]  Henning Stichtenoth,et al.  Note on Niederreiter-Xing's Propagation Rule for Linear Codes , 2002, Applicable Algebra in Engineering, Communication and Computing.

[10]  Patrick Fitzpatrick,et al.  Algebraic structure of quasicyclic codes , 2001, Discret. Appl. Math..

[11]  Graham H. Norton,et al.  Matrix-Product Codes over ?q , 2001, Applicable Algebra in Engineering, Communication and Computing.

[12]  Venkatesan Guruswami,et al.  Better Binary List Decodable Codes Via Multilevel Concatenation , 2007, IEEE Transactions on Information Theory.

[13]  Peter Elias,et al.  List decoding for noisy channels , 1957 .

[14]  Vladimir M. Blinovsky,et al.  List decoding , 1992, Discret. Math..