8 research outputs found
On the weight distributions of several classes of cyclic codes from APN monomials
Let be an odd integer and be an odd prime. % with ,
where is an odd integer.
In this paper, many classes of three-weight cyclic codes over
are presented via an examination of the condition for the
cyclic codes and , which have
parity-check polynomials and respectively, to
have the same weight distribution, where is the minimal polynomial of
over for a primitive element of
. %For , the duals of five classes of the proposed
cyclic codes are optimal in the sense that they meet certain bounds on linear
codes. Furthermore, for and positive integers such
that there exist integers with and satisfying , the value
distributions of the two exponential sums T(a,b)=\sum\limits_{x\in
\mathbb{F}_{p^m}}\omega^{\Tr(ax+bx^e)} and S(a,b,c)=\sum\limits_{x\in
\mathbb{F}_{p^m}}\omega^{\Tr(ax+bx^e+cx^s)}, where , are
settled. As an application, the value distribution of is utilized to
investigate the weight distribution of the cyclic codes
with parity-check polynomial . In the case of and
even satisfying the above condition, the duals of the cyclic codes
have the optimal 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
The Subfield Codes of Some Few-Weight Linear Codes
Subfield codes of linear codes over finite fields have recently received a
lot of attention, as some of these codes are optimal and have applications in
secrete sharing, authentication codes and association schemes. In this paper,
the -ary subfield codes of six different families of
linear codes are presented, respectively. The parameters and
weight distribution of the subfield codes and their punctured codes
are explicitly determined. The parameters of the duals of
these codes are also studied. Some of the resultant -ary codes
and their dual codes are optimal
and some have the best known parameters. The parameters and weight enumerators
of the first two families of linear codes are also settled,
among which the first family is an optimal two-weight linear code meeting the
Griesmer bound, and the dual codes of these two families are almost MDS codes.
As a byproduct of this paper, a family of quaternary
Hermitian self-dual code are obtained with . As an application,
several infinite families of 2-designs and 3-designs are also constructed with
three families of linear codes of this paper.Comment: arXiv admin note: text overlap with arXiv:1804.06003,
arXiv:2207.07262 by other author