2,920 research outputs found
Five Families of Three-Weight Ternary Cyclic Codes and Their Duals
As a subclass of linear codes, cyclic codes have applications in consumer
electronics, data storage systems, and communication systems as they have
efficient encoding and decoding algorithms. In this paper, five families of
three-weight ternary cyclic codes whose duals have two zeros are presented. The
weight distributions of the five families of cyclic codes are settled. The
duals of two families of the cyclic codes are optimal
Construction of quasi-cyclic self-dual codes
There is a one-to-one correspondence between -quasi-cyclic codes over a
finite field and linear codes over a ring . Using this correspondence, we prove that every
-quasi-cyclic self-dual code of length over a finite field
can be obtained by the {\it building-up} construction, provided
that char or , is a prime , and
is a primitive element of . We determine possible weight
enumerators of a binary -quasi-cyclic self-dual code of length
(with a prime) in terms of divisibility by . We improve the result of
[3] by constructing new binary cubic (i.e., -quasi-cyclic codes of length
) optimal self-dual codes of lengths (Type I), 54 and
66. We also find quasi-cyclic optimal self-dual codes of lengths 40, 50, and
60. When , we obtain a new 8-quasi-cyclic self-dual code
over and a new 6-quasi-cyclic self-dual code over
. When , we find a new 4-quasi-cyclic self-dual
code over and a new 6-quasi-cyclic self-dual code
over .Comment: 25 pages, 2 tables; Finite Fields and Their Applications, 201
Four infinite families of ternary cyclic codes with a square-root-like lower bound
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. Inspired by the recent work on binary cyclic
codes published in IEEE Trans. Inf. Theory, vol. 68, no. 12, pp. 7842-7849,
2022, and the arXiv paper arXiv:2301.06446, the objectives of this paper are
the construction and analyses of four infinite families of ternary cyclic codes
with length for odd and dimension
whose minimum distances have a square-root-like lower bound. Their duals have
parameters , where and
also has a square-root-like lower bound. These families of codes and
their duals contain distance-optimal cyclic codes
Codes and Pseudo-Geometric Designs from the Ternary -Sequences with Welch-type decimation
Pseudo-geometric designs are combinatorial designs which share the same
parameters as a finite geometry design, but which are not isomorphic to that
design. As far as we know, many pseudo-geometric designs have been constructed
by the methods of finite geometries and combinatorics. However, none of
pseudo-geometric designs with the parameters is constructed by the approach of coding theory. In
this paper, we use cyclic codes to construct pseudo-geometric designs. We
firstly present a family of ternary cyclic codes from the -sequences with
Welch-type decimation , and obtain some infinite family
of 2-designs and a family of Steiner systems
using these cyclic codes and their duals. Moreover, the parameters of these
cyclic codes and their shortened codes are also determined. Some of those
ternary codes are optimal or almost optimal. Finally, we show that one of these
obtained Steiner systems is inequivalent to the point-line design of the
projective space and thus is a pseudo-geometric design.Comment: 15 pages. arXiv admin note: text overlap with arXiv:2206.15153,
arXiv:2110.0388
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