A characterization of MDS codes that have an error correcting pair

Error-correcting pairs were introduced in 1988 by R. Pellikaan, and were found independently by R. K\"otter (1992), as a general algebraic method of decoding linear codes. These pairs exist for several classes of codes. However little or no study has been made for characterizing those codes. This article is an attempt to fill the vacuum left by the literature concerning this subject. Since every linear code is contained in an MDS code of the same minimum distance over some finite field extension we have focused our study on the class of MDS codes. Our main result states that an MDS code of minimum distance $2t+1$ has a $t$-ECP if and only if it is a generalized Reed-Solomon code. A second proof is given using recent results Mirandola and Z\'emor (2015) on the Schur product of codes

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

Some constructions of quantum MDS codes

We construct quantum MDS codes with parameters $[\![ q^2+1,q^2+3-2d,d ]\!] _q$ for all $d \leqslant q+1$, $d \neq q$. These codes are shown to exist by proving that there are classical generalised Reed-Solomon codes which contain their Hermitian dual. These constructions include many constructions which were previously known but in some cases these codes are new. We go on to prove that if $d\geqslant q+2$ then there is no generalised Reed-Solomon $[n,n-d+1,d]_{q^2}$ code which contains its Hermitian dual. We also construct an $[\![ 18,0,10 ]\!] _5$ quantum MDS code, an $[\![ 18,0,10 ]\!] _7$ quantum MDS code and a $[\![ 14,0,8 ]\!] _5$ quantum MDS code, which are the first quantum MDS codes discovered for which $d \geqslant q+3$, apart from the $[\![ 10,0,6 ]\!] _3$ quantum MDS code derived from Glynn's code

Some constructions of quantum MDS codes

​The version of record os available online at: http://dx.doi.org/10.1007/s10623-021-00846-yWe construct quantum MDS codes with parameters [[q2+1,q2+3-2d,d]]q for all d¿q+1, d¿q. These codes are shown to exist by proving that there are classical generalised Reed–Solomon codes which contain their Hermitian dual. These constructions include many constructions which were previously known but in some cases these codes are new. We go on to prove that if d¿q+2 then there is no generalised Reed–Solomon [n,n-d+1,d]q2 code which contains its Hermitian dual. We also construct an [[18,0,10]]5 quantum MDS code, an [[18,0,10]]7 quantum MDS code and a [[14,0,8]]5 quantum MDS code, which are the first quantum MDS codes discovered for which d¿q+3, apart from the [[10,0,6]]3 quantum MDS code derived from Glynn’s code.Postprint (author's final draft

Structural Properties of Twisted Reed-Solomon Codes with Applications to Cryptography

We present a generalisation of Twisted Reed-Solomon codes containing a new large class of MDS codes. We prove that the code class contains a large subfamily that is closed under duality. Furthermore, we study the Schur squares of the new codes and show that their dimension is often large. Using these structural properties, we single out a subfamily of the new codes which could be considered for code-based cryptography: These codes resist some existing structural attacks for Reed-Solomon-like codes, i.e. methods for retrieving the code parameters from an obfuscated generator matrix.Comment: 5 pages, accepted at: IEEE International Symposium on Information Theory 201