50 research outputs found
Several families of ternary negacyclic codes and their duals
Constacyclic codes contain cyclic codes as a subclass and have nice algebraic
structures. Constacyclic codes have theoretical importance, as they are
connected to a number of areas of mathematics and outperform cyclic codes in
several aspects. Negacyclic codes are a subclass of constacyclic codes and are
distance-optimal in many cases. However, compared with the extensive study of
cyclic codes, negacyclic codes are much less studied. In this paper, several
families of ternary negacyclic codes and their duals are constructed and
analysed. These families of negacyclic codes and their duals contain
distance-optimal codes and have very good parameters in general
Two types of negacyclic BCH codes
Negacyclic BCH codes are an important subclass of negacyclic codes, which
have efficient encoding and decoding algorithms, but their parameters are
difficult to determine. In this paper, we mainly study two types of negacyclic
BCH codes of length . As byproducts, we
investigate the first three largest odd coset leaders modulo . The
parameters of two types of negacyclic BCH codes are analysed with small and
large dimensions, and the weight distribution of neagcyclic BCH codes of length
are determined for designed distance in some ranges
New bounds for -Symbol Distances of Matrix Product Codes
Matrix product codes are generalizations of some well-known constructions of
codes, such as Reed-Muller codes, -construction, etc. Recently, a
bound for the symbol-pair distance of a matrix product code was given in
\cite{LEL}, and new families of MDS symbol-pair codes were constructed by using
this bound. In this paper, we generalize this bound to the -symbol distance
of a matrix product code and determine all minimum -symbol distances of
Reed-Muller codes. We also give a bound for the minimum -symbol distance of
codes obtained from the -construction, and use this bound to
construct some -linear -symbol almost MDS codes with arbitrary
length. All the minimum -symbol distances of -linear codes and
-linear codes for are determined. Some examples are
presented to illustrate these results
Two classes of reducible cyclic codes with large minimum symbol-pair distances
The high-density data storage technology aims to design high-capacity storage
at a relatively low cost. In order to achieve this goal, symbol-pair codes were
proposed by Cassuto and Blaum \cite{CB10,CB11} to handle channels that output
pairs of overlapping symbols. Such a channel is called symbol-pair read
channel, which introduce new concept called symbol-pair weight and minimum
symbol-pair distance. In this paper, we consider the parameters of two classes
of reducible cyclic codes under the symbol-pair metric. Based on the theory of
cyclotomic numbers and Gaussian period over finite fields, we show the possible
symbol-pair weights of these codes. Their minimum symbol-pair distances are
twice the minimum Hamming distances under some conditions. Moreover, we obtain
some three symbol-pair weight codes and determine their symbol-pair weight
distribution. A class of MDS symbol-pair codes is also established. Among other
results, we determine the values of some generalized cyclotomic numbers
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
Infinite families of cyclic and negacyclic codes supporting 3-designs
Interplay between coding theory and combinatorial -designs has been a hot
topic for many years for combinatorialists and coding theorists. Some infinite
families of cyclic codes supporting infinite families of -designs have been
constructed in the past 50 years. However, no infinite family of negacyclic
codes supporting an infinite family of -designs has been reported in the
literature. This is the main motivation of this paper. Let , where
is an odd prime and is an integer. The objective of this paper is to
present an infinite family of cyclic codes over \gf(q) supporting an infinite
family of -designs and two infinite families of negacyclic codes over
\gf(q^2) supporting two infinite families of -designs. The parameters and
the weight distributions of these codes are determined. The subfield subcodes
of these negacyclic codes over \gf(q) are studied. Three infinite families of
almost MDS codes are also presented. A constacyclic code over GF()
supporting a -design and six open problems are also presented in this paper
A transform approach to polycyclic and serial codes over rings
Producción CientíficaIn this paper, a transform approach is used for polycyclic and serial codes over finite local rings in the case that the defining polynomials have no multiple roots. This allows us to study them in terms of linear algebra and invariant subspaces as well as understand the duality in terms of the transform domain. We also make a characterization of when two polycyclic ambient spaces are Hamming-isometric.Ministerio de Ciencia, Innovación y Universidades / Agencia Estatal de Investigación / 0.13039/501100011033 (grant PGC2018-096446-B-C21
Several Families of Irreducible Constacyclic and Cyclic Codes
In this paper, several families of irreducible constacyclic codes over finite
fields and their duals are studied. The weight distributions of these
irreducible constacyclic codes and the parameters of their duals are settled.
Several families of irreducible constacyclic codes with a few weights and
several families of optimal constacyclic codes are constructed. As by-products,
a family of irreducible cyclic codes over
\gf(q) and a family of
irreducible cyclic codes over \gf(q) are presented, where is a prime such
that \ord_n(q)=(n-1)/2. The results in this paper complement earlier works on
irreducible constacyclic and cyclic codes over finite fields