Group rings, G-codes and constructions of self-dual and formally self-dual codes

We describe G-codes, which are codes that are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a G-code is also a G-code. We give constructions of self-dual and formally self-dual codes in this setting and we improve the existing construction given in Hurley (Int J Pure Appl Math 31(3):319–335, 2006) by showing that one of the conditions given in the theorem is unnecessary and, moreover, it restricts the number of self-dual codes obtained by the construction. We show that several of the standard constructions of self-dual codes are found within our general framework. We prove that our constructed codes must have an automorphism group that contains G as a subgroup. We also prove that a common construction technique for producing self-dual codes cannot produce the putative [72, 36, 16] Type II code. Additionally, we show precisely which groups can be used to construct the extremal Type II codes over length 24 and 48. We define quasi-G codes and give a construction of these codes.

[1]  Steven T. Dougherty,et al.  Cyclic codes over Rk , 2012, Des. Codes Cryptogr..

[2]  Patrick Solé,et al.  Cyclic codes over M2(F2) , 2013, J. Frankl. Inst..

[3]  L. Thiel,et al.  The Non-Existence of Finite Projective Planes of Order 10 , 1989, Canadian Journal of Mathematics.

[4]  Wolfgang Willems,et al.  On the Automorphism Group of a Binary Self-Dual Doubly Even $[{72, 36, 16}]$ Code , 2011, IEEE Transactions on Information Theory.

[5]  Member INdAM-GNSAGA.  The automorphism group of a self-dual [ 72 , 36 , 16 ] code is not an elementary abelian group of order 8 , 2013 .

[6]  Ian McLoughlin.  A group ring construction of the [48,24,12] type II linear block code , 2012, Des. Codes Cryptogr..

[7]  Martino Borello.  The automorphism group of a self-dual [72,36,16] code is not an elementary abelian group of order 8 , 2013, ArXiv.

[8]  Gabriele Nebe.  An extremal [72, 36, 16] binary code has no automorphism group containing Z2 x Z4, Q8, or Z10 , 2012, Finite Fields Their Appl..

[9]  Nikolay Yankov.  A Putative Doubly Even [72,36,16] Code Does Not Have an Automorphism of Order 9 , 2012, IEEE Transactions on Information Theory.

[10]  Steven T. Dougherty,et al.  Codes over Rk, Gray maps and their binary images , 2011, Finite Fields Their Appl..

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

[12]  S. Dougherty,et al.  Self-Dual Codes over R_k and Binary Self-Dual Codes , 2013 .

[13]  S. Dougherty,et al.  Algebraic Coding Theory Over Finite Commutative Rings , 2017 .

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

[15]  Steven T. Dougherty,et al.  Constructions of self-dual codes and formally self-dual codes over rings , 2016, Applicable Algebra in Engineering, Communication and Computing.

[16]  Stefka Bouyuklieva,et al.  The Automorphism Group of a Binary Self-Dual Doubly Even [72, 36, 16] Code is Solvable , 2006, IEEE Trans. Inf. Theory.

[17]  Stefka Bouyuklieva.  On the Automorphisms of Order 2 with Fixed Points for the Extremal Self-Dual Codes of Length 24"m , 2002, Des. Codes Cryptogr..

[18]  F. J. Mac Williams.  Binary codes which are ideals in the group algebra of an abelian group , 1970 .

[19]  T. Hurley,et al.  Group rings and rings of matrices , 2006 .

[20]  Steven T. Dougherty,et al.  Quasi-cyclic codes as cyclic codes over a family of local rings , 2016, Finite Fields Their Appl..