Abstract
In this work, we apply the idea of composite matrices arising from group rings to derive a number of different techniques for constructing self-dual codes over finite commutative Frobenius rings. By applying these techniques over different alphabets, we construct best known singly-even binary self-dual codes of lengths 80, 84 and 96 as well as doubly-even binary self-dual codes of length 96 that were not known in the literature before.
Highlights
Self-dual codes form a family of widely studied linear codes which have many interesting properties and are intimately connected with many mathematical structures such as designs, lattices, modular forms and sphere packings
The most famous of these techniques is quite possibly the pure double circulant construction, which utilises a generator matrix of the form G = (In | A) where In is the n × n identity matrix and A is an n × n circulant matrix. It follows that G is a generator matrix of a self-dual [2n, n] code if and only if A AT = −In. This technique has since been generalised by assuming a generator matrix of the form G = (In | σ (v)) where σ is an isomorphism
Using generator matrices of the form (In | Ω(v)) for a number of different composite matrices Ω(v), we find many self-dual codes with weight enumerator parameters of previously unknown values
Summary
Self-dual codes form a family of widely studied linear codes which have many interesting properties and are intimately connected with many mathematical structures such as designs, lattices, modular forms and sphere packings. It clearly follows that (In | Ω(v)) is a generator matrix of a self-dual code if and only if Ω(v)Ω(v)T = −In. The idea of composite matrices was first introduced in [11] as a way of generalising the structure of σ (v). The main problem we face when attempting to construct codes with such a generator matrix is choosing parameters in such a way that allows for structural complexity of Ω(v) while allowing for a reasonable set of necessary and sufficient conditions for the satisfaction of Ω(v)Ω(v)T = −In. Using generator matrices of the form (In | Ω(v)) for a number of different composite matrices Ω(v), we find many self-dual codes with weight enumerator parameters of previously unknown values (relative to referenced sources). We finish with concluding remarks and discussion of possible expansion on this work
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