An Algorithm for Computing the Minimum Distances of Extensions of BCH Codes Embedded in Semigroup Rings

An algorithm is given for computing the weights of extensions of BCH codes embedded in semigroup rings as ideals. The algorithm relies on a more general technical result of independent interest.

[1]  Mohammad Umar Siddiqi,et al.  Transform domain characterization of Abelian codes , 1992, IEEE Trans. Inf. Theory.

[2]  R. Lidl,et al.  Applied abstract algebra , 1984 .

[3]  The radical of the algebra of any finite semigroup over any field , 1970 .

[4]  Roberta Evans Sabin,et al.  On minimum distance bounds for abelian codes , 1992, Applicable Algebra in Engineering, Communication and Computing.

[5]  A. Quesada,et al.  On semisimple semigroup rings , 1980 .

[6]  Andrei V. Kelarev.  Minimum distances and information rates for matrix extensions of BCH codes , 2004 .

[7]  S. Berman.  On the theory of group codes , 1967 .

[8]  A. Kelarev.  Radicals of semigroup rings of commutative semigroups , 1994 .

[9]  Charles C. Sims,et al.  Computation with finitely presented groups , 1994, Encyclopedia of mathematics and its applications.

[10]  Pedro A. García-Sánchez,et al.  Finitely generated commutative monoids , 1999 .

[11]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[12]  Michael Francis Atiyah,et al.  Introduction to commutative algebra , 1969 .

[13]  Rudolf Lide,et al.  Finite fields , 1983 .

[14]  E. F. Robertson,et al.  Reidemeister-Schreier type rewriting for semigroups , 1995 .

[15]  C. Rentería,et al.  Reed‐muller codes: an ideal theory approach , 1997 .

[16]  William Stallings,et al.  Wireless communications and networking , 2002 .

[17]  Andrei V. Kelarev,et al.  Generators and weights of polynomial codes , 1997 .

[18]  Lev N. Shevrin,et al.  Semigroups and their subsemigroup lattices , 1996, Mathematics and its applications.

[19]  T. E. Hall.  Biprefix Codes, Inverse Semigroups and Syntactic Monoids of Injective Automata , 1984, Theor. Comput. Sci..

[20]  A. Kelarev.  Graph algebras and automata , 2003 .

[21]  Olaf Manz,et al.  Classical codes as ideals in group algebras , 1992, Des. Codes Cryptogr..

[22]  Andrei V. Kelarev,et al.  Information Rates and Weights of Codes in Structural Matrix Rings , 2001, AAECC.

[23]  A. Kelarev,et al.  On cyclic codes in incidence rings , 2006 .

[24]  Andrei V. Kelarev,et al.  A Polynomial Algorithm for Codes Based on Directed Graphs , 2006, CATS.

[25]  A. Kelarev.  Ring constructions and applications , 2002 .

[26]  P. A. Grillet.  Semigroups: An Introduction to the Structure Theory , 1995 .

[27]  Nik Ruskuc.  Presentations for Subgroups of Monoids , 1999 .

[28]  D. Vertigan.  Latroids and their representation by codes over modules , 2003 .