Group codes over fields are asymptotically good

Group codes are right or left ideals in a group algebra of a finite group over a finite field. Following ideas of Bazzi and Mitter on group codes over the binary field, we prove that group codes over finite fields of any characteristic are asymptotically good.

[1]  Jonathan L. Alperin,et al.  Groups and Representations , 1995 .

[2]  Steven Roman,et al.  Coding and information theory , 1992 .

[3]  P. Piret An upper bound on the weight distribution of some systematic codes , 1985, IEEE Trans. Inf. Theory.

[4]  Olaf Manz,et al.  The extended golay codes considered as ideals , 1990, J. Comb. Theory, Ser. A.

[5]  Primes in a prescribed arithmetic progression dividing the sequence a^k+b^k , 2007, math/0703925.

[6]  Hongwei Liu,et al.  Checkable Codes from Group Rings , 2010, ArXiv.

[7]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[8]  P. Charpin Une généralisation de la construction de Berman des codes de Reed et Muller p-aires , 1988 .

[9]  Martino Borello,et al.  On checkable codes in group algebras , 2019, ArXiv.

[10]  James L. Massey On the fractional weight of distinct binary n -tuples (Corresp.) , 1974, IEEE Trans. Inf. Theory.

[11]  Patrick Solé,et al.  Asymptotic performance of metacyclic codes , 2020, Discret. Math..

[12]  Sanjoy K. Mitter,et al.  Some randomized code constructions from group actions , 2006, IEEE Transactions on Information Theory.

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

[14]  Ian McLoughlin,et al.  A Group Ring Construction of the Extended Binary Golay Code , 2008, IEEE Transactions on Information Theory.

[15]  B. Huppert,et al.  Finite Groups II , 1982 .

[16]  Yun Fan,et al.  Thresholds of Random Quasi-Abelian Codes , 2013, IEEE Transactions on Information Theory.

[17]  Wolfgang Willems,et al.  Codes of Small Defect , 1997, Des. Codes Cryptogr..