New self-dual codes of length 68 from a $ 2 \times 2 $ block matrix construction and group rings

Abstract

Many generator matrices for constructing extremal binary selfdual codes of different lengths have the form G = (In | A), where In is the n×n identity matrix and A is the n×n matrix fully determined by the first row. In this work, we define a generator matrix in which A is a block matrix, where the blocks come from group rings and also, A is not fully determined by the elements appearing in the first row. By applying our construction over F 2 +uF 2 and by employing the extension method for codes, we were able to construct new extremal binary self-dual codes of length 68. Additionally, by employing a generalised neighbour method to the codes obtained, we were able to construct many new binary self-dual [68, 34, 12]-codes with the rare parameters γ = 7, 8 and 9 in W 68,2. In particular, we find 92 new binary self-dual [68, 34, 12]-codes.

New binary self-dual codes of lengths 80, 84 and 96 from composite matrices

“…Assume A is an n × n matrix over a commutative ring R where n ≥ 2. Suppose we decompose A into blocks such that 1) . Then by block-wise multiplication we obtain…”

“…The isomorphism σ is such that G is a generator matrix of a self-dual [2n, n] code if and only if v is a unitary unit in the group ring. See [15,7,1,14] for recent applications of this isomorphism in constructing self-dual codes.…”

New binary self-dual codes of lengths 80, 84 and 96 from composite matrices

“…2.4 for more details). Recent applications of this isomorphism in constructing self-dual codes can be seen in [1,8,15,16].…”

Quaternary Hermitian self-dual codes of lengths 26, 32, 36, 38 and 40 from modifications of well-known circulant constructions

“…When searching for n × n matrices to use in constructions of (Hermitian) self-dual codes, by assuming these matrices are circulant we reduce the size of the search field from n 2 to n. For this reason, circulant matrices have been used extensively to construct (Hermitian) self-dual codes. See [11,7,12,2,13,10,9] for recent utilisation of circulant matrices in constructing self-dual codes. In this work, we give three different modifications of various well-known circulant constructions of self-dual codes, which we apply to construct optimal and best known quaternary Hermitian self-dual codes.…”