7 research outputs found
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
Further results on several classes of optimal ternary cyclic codes with minimum distance four
Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, by analyzing the solutions of certain equations over and using the multivariate method, we present three classes of optimal ternary cyclic codes in the case of is odd and five classes of optimal ternary cyclic codes with explicit values , respectively. In addition, two classes of optimal ternary cyclic codes are given