10 research outputs found
Constacyclic Codes over Finite Fields
An equivalence relation called isometry is introduced to classify
constacyclic codes over a finite field; the polynomial generators of
constacyclic codes of length are characterized, where is the
characteristic of the finite field and is a prime different from
Recent progress on weight distributions of cyclic codes over finite fields
Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. In coding theory it is often desirable to know the weight distribution of a cyclic code to estimate the error correcting capability and error probability. In this paper, we present the recent progress on the weight distributions of cyclic codes over finite fields, which had been determined by exponential sums. The cyclic codes with few weights which are very useful are discussed and their existence conditions are listed. Furthermore, we discuss the more general case of constacyclic codes and give some equivalences to characterize their weight distributions
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
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
Twisted skew -codes
In this paper we investigate left ideals as codes in twisted skew group
rings. The considered rings, which are often algebras over a finite field,
allows us to detect many of the well-known codes. The presentation, given here,
unifies the concept of group codes, twisted group codes and skew group codes
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