Constructions for self-dual codes induced from group rings

Abstract

In this work, we establish a strong connection between group rings and self-dual codes. We prove that a group ring element corresponds to a self-dual code if and only if it is a unitary unit. We also show that the double-circulant and four-circulant 1 constructions come from cyclic and dihedral groups, respectively. Using groups of order 8 and 16 we find many new construction methods, in addition to the well-known methods, for self-dual codes. We establish the relevance of these new constructions by finding many extremal binary self-dual codes using them, which we list in several tables. In particular, we construct 10 new extremal binary self-dual codes of length 68.

Bordered constructions of self-dual codes from group rings and new extremal binary self-dual codes

“…The groups of different orders have introduced many new constructions that lead to different automorphism groups, which in turn fill the gap caused by the restrictive nature of constructions such as double circulant, bordered double circulant and four circulant constructions. The success of these new methods, which has been demonstrated in [13] and [12], is further shown to be the case for the methods we have introduced in this paper, which modify and extend the concept of bordered double circulant matrices as well as the classical group ring constructions that have been discussed in the aforementioned works. The remarkable number of extremal self-dual codes and the 41 new extremal self-dual codes of length 68 that have been obtained show the strength of the methods we have discussed.Due to computational limitations and page restrictions, we have considered a sample of groups in our constructions.…”

“…The main idea in the most recent works is to use a matrix of the form [I n |A], where A is the image of a unitary unit in a group ring under a map that sends group ring elements to matrices. This idea has been successfully used in producing extremal binary self-dual codes in [12] and [13].…”

“…In this work, we will modify the constructions by introducing a border to I n as well as to the matrix A used in the group ring constructions found in [12] and [13]. Using groups of orders, 9,15,21,25,27,33 and 35 over the binary field or the ring R 1 = F 2 + uF 2 , we construct a considerable number of extremal binary self-dual codes of lengths 20, 32, 40, 44, 52, 56, 64, 68, 88 and best known binary self-dual codes of length 72.…”

Quaternary group ring codes: Ranks, kernels and self-dual codes

“…This construction was given for codes over rings in [8] and for codes over fields in [14]. It was used in [11] to construct self-dual codes. Let v = v 1 g 1 + · · · + v n g n ∈ Z 4 G. Notice that the elements g −1 1 , g −1 2 , .…”

An altered four circulant construction for self-dual codes from group rings and new extremal binary self-dual codes I

“…(1, 0) (0, 0, 1, 0, 0, 1, 1, 1, 1) (1, 1) (0, 0, 1, 0, 1, 1, 1, 0, 1) 2 3 · 3 2 · 5 · 19 [40, 20, 8] II C 3 × C 3 (0, 0) (0, 0, 0, 0, 1, 1, 0, 1, 1) (0, 1) (0, 0, 1, 0, 0, 1, 1, 1, 0) 2 15 · 3 2 · 5 [40, 20, 8] I C 3 × C 3 (1, 0) (0, 0, 0, 0, 0, 0, 1, 1, 1) (1, 1) (0, 0, 1, 0, 0, 1, 0, 1, 0) 2 4 · 3 4 [40, 20, 8] II C 3 × C 3 (1, 0) (0, 0, 1, 0, 0, 1, 1, 1, 1) (1, 1) (0, 0, 1, 1, 1, 0, 1, 1, 0) 2 15 40,14] code was constructed in [18] with α = −280, β = 10. In [27], [80,40,14] codes were constructed for β = 0 and α = −17k where k ∈ {2, .…”