43 research outputs found
On MDS Negacyclic LCD Codes
Linear codes with complementary duals (LCD) have a great deal of significance
amongst linear codes. Maximum distance separable (MDS) codes are also an
important class of linear codes since they achieve the greatest error
correcting and detecting capabilities for fixed length and dimension. The
construction of linear codes that are both LCD and MDS is a hard task in coding
theory. In this paper, we study the constructions of LCD codes that are MDS
from negacyclic codes over finite fields of odd prime power elements. We
construct four families of MDS negacyclic LCD codes of length
, and a family of negacyclic LCD codes
of length . Furthermore, we obtain five families of -ary
Hermitian MDS negacyclic LCD codes of length and four
families of Hermitian negacyclic LCD codes of length For both
Euclidean and Hermitian cases the dimensions of these codes are determined and
for some classes the minimum distances are settled. For the other cases, by
studying and -cyclotomic classes we give lower bounds on the minimum
distance
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
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
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
Weight distribution of a class of cyclic codes of length
Let be a finite field with elements and be a positive integer. In this paper, we determine the weight distribution of a class cyclic codes of length over whose parity check polynomials are either binomials or trinomials with zeros over , where integer . In addition, constant weight and two-weight linear codes are constructed when