Construction for both self-dual codes and LCD codes

<p style='text-indent:20px;'>From a given [<i>n</i>, <i>k</i>] code <i>C</i>, we give a method for constructing many [<i>n</i>, <i>k</i>] codes <i>C</i>^' such that the hull dimensions of <i>C</i> and <i>C</i>^' are identical. This method can be applied to constructions of both self-dual codes and linear complementary dual codes (LCD codes for short). Using the method, we construct 661 new inequivalent extremal doubly even [56, 28, 12] codes. Furthermore, constructing LCD codes by the method, we improve some of the previously known lower bounds on the largest minimum weights of binary LCD codes of length 26 ≤ <i>n</i> ≤ 40.</p>

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

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.

Skew G-codes

We describe skew [Formula: see text]-codes, which are codes that are the ideals in a skew group ring, where the ring is a finite commutative Frobenius ring and [Formula: see text] is an arbitrary finite group. These codes generalize many of the well-known classes of codes such as cyclic, quasicyclic, constacyclic codes, skew cyclic, skew quasicyclic and skew constacyclic codes. Additionally, using the skew [Formula: see text]-matrices, we can generalize almost all the known constructions in the literature for self-dual codes.

Constacyclic Codes over Finite Chain Rings of Characteristic p

Let R be a finite commutative chain ring of characteristic p with invariants p,r, and k. In this paper, we study λ-constacyclic codes of an arbitrary length N over R, where λ is a unit of R. We first reduce this to investigate constacyclic codes of length ps (N=n1ps,p∤n1) over a certain finite chain ring CR(uk,rb) of characteristic p, which is an extension of R. Then we use discrete Fourier transform (DFT) to construct an isomorphism γ between R[x]/<xN−λ> and a direct sum ⊕b∈IS(rb) of certain local rings, where I is the complete set of representatives of p-cyclotomic cosets modulo n1. By this isomorphism, all codes over R and their dual codes are obtained from the ideals of S(rb). In addition, we determine explicitly the inverse of γ so that the unique polynomial representations of λ-constacyclic codes may be calculated. Finally, for k=2 the exact number of such codes is provided.