74 research outputs found
On Hull-Variation Problem of Equivalent Linear Codes
The intersection () of a linear code and its Euclidean dual (Hermitian dual ) is called the Euclidean
(Hermitian) hull of this code. The construction of an entanglement-assisted
quantum code from a linear code over or depends
essentially on the Euclidean hull or the Hermitian hull of this code. Therefore
it is natural to consider the hull-variation problem when a linear code is transformed to an equivalent code . In this paper
we introduce the maximal hull dimension as an invariant of a linear code with
respect to the equivalent transformations. Then some basic properties of the
maximal hull dimension are studied. A general method to construct
hull-decreasing or hull-increasing equivalent linear codes is proposed. We
prove that for a nonnegative integer satisfying , a
linear self-dual code is equivalent to a linear -dimension hull
code. On the opposite direction we prove that a linear LCD code over satisfying and is equivalent to a linear
one-dimension hull code under a weak condition. Several new families of
negacyclic LCD codes and BCH LCD codes over are also constructed.
Our method can be applied to the generalized Reed-Solomon codes and the
generalized twisted Reed-Solomon codes to construct arbitrary dimension hull
MDS codes. Some new EAQEC codes including MDS and almost MDS
entanglement-assisted quantum codes are constructed. Many EAQEC codes over
small fields are constructed from optimal Hermitian self-dual codes.Comment: 33 pages, minor error correcte
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 for any given
small consecutive lengths , which are not equivalent to Reed-Solomon codes
and twisted Reed-Solomon codes. 2) New self-dual MDS AG codes over 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
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