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

Abstract

We give constructions of self-dual and formally self-dual codes from group rings where the ring is a finite commutative Frobenius ring. We improve the existing construction given in [11] 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 correspond to ideals in the group ring RG and as such 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.

Constructing self-dual codes from group rings and reverse circulant matrices

“…The following construction of a matrix was first given for codes over fields by Hurley in [21]. It was extended to Frobenius rings in [10]. Let R be a finite commutative Frobenius ring and let G = {g 1 , g 2 , .…”

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

“…We now give a construction of a code from an element of the group ring. 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.…”

“…It is proven in [8] that if I(C) is an ideal in Z 4 G then I(C ⊥ ) is an ideal in Z 4 G. The following gives an example where the kernel of a group code over Z 4 is a minimum for all groups G. Specifically, where K(C(v)) = C(2v).…”

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

“…A classical way to produce a code from a group ring is to use the matrix representation of elements in a group ring and using zero divisors ( [16], [17]). In [6], the idea was extended to any group G and G-codes were defined as codes that are ideals in the group ring RG where R is a finite Frobenius ring. Group rings have also been used in constructing extremal binary self-dual codes.…”