A Modified Bordered Construction for Self-Dual Codes from Group Rings

We describe a bordered construction for self-dual codes coming from group rings. We apply the constructions coming from the cyclic and dihedral groups over several alphabets to obtain extremal binary self-dual codes of various lengths. In particular we find a new extremal binary self-dual code of length 78

New extremal binary self-dual codes of length 68 via short kharaghani array over f_2 + uf_2

In this work, new construction methods for self-dual codes are given. The methods use the short Kharaghani array and a variation of it. These are applicable to any commutative Frobenius ring. We apply the constructions over the ring F_2 + uF_2 and self-dual Type I [64, 32, 12]_2-codes with various weight enumerators obtained as Gray images. By the use of an extension theorem for self-dual codes we were able to construct 27 new extremal binary self-dual codes of length 68. The existence of the extremal binary self-dual codes with these weight enumerators was previously unknown.Comment: 10 pages, 5 table

Euclidean and Hermitian LCD MDS codes

Linear codes with complementary duals (abbreviated LCD) are linear codes whose intersection with their dual is trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Non-binary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. On the other hand, being optimal codes, maximum distance separable codes (abbreviated MDS) have been of much interest from many researchers due to their theoretical significant and practical implications. However, little work has been done on LCD MDS codes. In particular, determining the existence of $q$-ary $[n,k]$ LCD MDS codes for various lengths $n$ and dimensions $k$ is a basic and interesting problem. In this paper, we firstly study the problem of the existence of $q$-ary $[n,k]$ LCD MDS codes and completely solve it for the Euclidean case. More specifically, we show that for $q>3$ there exists a $q$-ary $[n,k]$ Euclidean LCD MDS code, where $0\le k \le n\le q+1$, or, $q=2^{m}$, $n=q+2$ and $k= 3 \text{or} q-1$. Secondly, we investigate several constructions of new Euclidean and Hermitian LCD MDS codes. Our main techniques in constructing Euclidean and Hermitian LCD MDS codes use some linear codes with small dimension or codimension, self-orthogonal codes and generalized Reed-Solomon codes

Construction of quasi-cyclic self-dual codes

There is a one-to-one correspondence between $\ell$-quasi-cyclic codes over a finite field $\mathbb F_q$ and linear codes over a ring $R = \mathbb F_q[Y]/(Y^m-1)$. Using this correspondence, we prove that every $\ell$-quasi-cyclic self-dual code of length $m\ell$ over a finite field $\mathbb F_q$ can be obtained by the {\it building-up} construction, provided that char $(\mathbb F_q)=2$ or $q \equiv 1 \pmod 4$, $m$ is a prime $p$, and $q$ is a primitive element of $\mathbb F_p$. We determine possible weight enumerators of a binary $\ell$-quasi-cyclic self-dual code of length $p\ell$ (with $p$ a prime) in terms of divisibility by $p$. We improve the result of [3] by constructing new binary cubic (i.e., $\ell$-quasi-cyclic codes of length $3\ell$) optimal self-dual codes of lengths $30, 36, 42, 48$ (Type I), 54 and 66. We also find quasi-cyclic optimal self-dual codes of lengths 40, 50, and 60. When $m=5$, we obtain a new 8-quasi-cyclic self-dual $[40, 20, 12]$ code over $\mathbb F_3$ and a new 6-quasi-cyclic self-dual $[30, 15, 10]$ code over $\mathbb F_4$. When $m=7$, we find a new 4-quasi-cyclic self-dual $[28, 14, 9]$ code over $\mathbb F_4$ and a new 6-quasi-cyclic self-dual $[42,21,12]$ code over $\mathbb F_4$.Comment: 25 pages, 2 tables; Finite Fields and Their Applications, 201

Duality Preserving Gray Maps for Codes over Rings

Given a finite ring $A$ which is a free left module over a subring $R$ of $A$, two types of $R$-bases, pseudo-self-dual bases (similar to trace orthogonal bases) and symmetric bases, are defined which in turn are used to define duality preserving maps from codes over $A$ to codes over $R$. Both types of bases are generalizations of similar concepts for fields. Many illustrative examples are given to shed light on the advantages to such mappings as well as their abundance

Self-Dual Codes

Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory. Topics covered in this survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems, bounds, mass formulae, enumeration, extremal codes, open problems. There is a comprehensive bibliography.Comment: 136 page