On the codes over the Z_3+vZ_3+v^2Z_3

TL;DR

This paper explores the structure and properties of various cyclic and skew codes over the ring Z_3+vZ_3+v^2Z_3, including their Gray images, duality, and applications to quantum error correction.

Contribution

It introduces new classes of quasi-constacyclic and skew quasi-constacyclic codes over the ring and provides conditions for their duality and freeness, with applications to quantum codes.

Findings

Gray images of codes are characterized.

Conditions for codes containing their duals are established.

Parameters for quantum error-correcting codes are derived.

Abstract

In this paper, we study the structure of cyclic, quasi-cyclic, constacyclic codes and their skew codes over the finite ring R=Z_3+vZ_3+v^2Z_3, v^3=v. The Gray images of cyclic, quasi-cyclic, skew cyclic, skew quasi-cyclic and skew constacyclic codes over R are obtained. A necessary and sufficient condition for cyclic (negacyclic) codes over R that contains its dual has been given. The parameters of quantum error correcting codes are obtained from both cyclic and negacyclic codes over R. It is given some examples. Firstly, quasi-constacyclic and skew quasi-constacyclic codes are introduced. By giving two product, it is investigated their duality. A sufficient condition for 1-generator skew quasi-constacyclic codes to be free is determined.