New Non-Equivalent (Self-Dual) MDS Codes From Elliptic Curves

It is well known that MDS codes can be constructed as algebraic geometric (AG) codes from elliptic curves. It is always interesting to construct new non-equivalent MDS codes and self-dual MDS codes. In recent years several constructions of new self-dual MDS codes from the generalized twisted Reed-Solomon codes were proposed. In this paper we construct new non-equivalent MDS and almost MDS codes from elliptic curve codes. 1) We show that there are many MDS AG codes from elliptic curves defined over ${\bf F}_q$ for any given small consecutive lengths $n$, which are not equivalent to Reed-Solomon codes and twisted Reed-Solomon codes. 2) New self-dual MDS AG codes over ${\bf F}_{{2^s}}$ from elliptic curves are constructed, which are not equivalent to Reed-Solomon codes and twisted Reed-Solomon codes. 3) Twisted versions of some elliptic curve codes are introduced such that new non-equivalent almost MDS codes are constructed. Moreover there are some non-equivalent MDS elliptic curve codes with the same length and the same dimension. The application to MDS entanglement-assisted quantum codes is given.We also construct non-equivalent new MDS codes of short lengths from higher genus curves.Comment: 28 pages, new non-equivalent MDS codes from higher genus curves are discusse

Infinite families of MDS and almost MDS codes from BCH codes

In this paper, the sufficient and necessary condition for the minimum distance of the BCH codes over $\mathbb{F}_q$ with length $q+1$ and designed distance 3 to be 3 and 4 are provided. Let $d$ be the minimum distance of the BCH code $\mathcal{C}_{(q,q+1,3,h)}$. We prove that (1) for any $q$, $d=3$ if and only if $\gcd(2h+1,q+1)>1$; (2) for $q$ odd, $d=4$ if and only if $\gcd(2h+1,q+1)=1$. By combining these conditions with the dimensions of these codes, the parameters of this BCH code are determined completely when $q$ is odd. Moreover, several infinite families of MDS and almost MDS (AMDS) codes are shown. Furthermore, the sufficient conditions for these AMDS codes to be distance-optimal and dimension-optimal locally repairable codes are presented. Based on these conditions, several examples are also given

New MDS self-dual codes over finite fields \F_{r^2}

MDS self-dual codes have nice algebraic structures and are uniquely determined by lengths. Recently, the construction of MDS self-dual codes of new lengths has become an important and hot issue in coding theory. In this paper, we develop the existing theory and construct six new classes of MDS self-dual codes. Together with our constructions, the proportion of all known MDS self-dual codes relative to possible MDS self-dual codes generally exceed 57\%. As far as we know, this is the largest known ratio. Moreover, some new families of MDS self-orthogonal codes and MDS almost self-dual codes are also constructed.Comment: 16 pages, 3 tabl