Sum-Rank BCH Codes and Cyclic-Skew-Cyclic Codes

by   Umberto Martínez-Peñas, et al.

In this work, cyclic-skew-cyclic codes and sum-rank BCH codes are introduced. Cyclic-skew-cyclic codes are characterized as left ideals over a suitable non-commutative finite ring, constructed using skew polynomials on top of polynomials (or vice versa). Single generators of such left ideals are found, and they are used to construct generator matrices of the corresponding codes. The notion of defining set is introduced, using pairs of roots of skew polynomials on top of poynomials. A lower bound (called sum-rank BCH bound) on the minimum sum-rank distance is given for cyclic-skew-cyclic codes whose defining set contains certain consecutive pairs. Sum-rank BCH codes, with prescribed minimum sum-rank distance, are then defined as the largest cyclic-skew-cyclic codes whose defining set contains such consecutive pairs. The defining set of a sum-rank BCH code is described, and a lower bound on its dimension is obtained. Thanks to it, tables are provided showing that sum-rank BCH codes beat previously known codes for the finite field of size 2^2 = 4 for the sum-rank metric for matrices with m=2 rows and columns. Finally, a decoder for sum-rank BCH codes up to half their prescribed distance is obtained.



There are no comments yet.


page 1

page 2

page 3

page 4


Sum-rank product codes and bounds on the minimum distance

The tensor product of one code endowed with the Hamming metric and one e...

Roos bound for skew cyclic codes in Hamming and rank metric

In this paper, a Roos like bound on the minimum distance for skew cyclic...

Self-Dual Skew Cyclic Codes over F_q+uF_q

In this paper, we give conditions for the existence of Hermitian self-du...

Cyclic subspace codes via the sum of Sidon spaces

Subspace codes, especially cyclic constant subspace codes, are of great ...

Maximum Sum-Rank Distance Codes over Finite Chain Rings

In this work, maximum sum-rank distance (MSRD) codes and linearized Reed...

Skew-Polynomial Rings and Skew-Cyclic Codes

This is a survey on the theory of skew-cyclic codes based on skew-polyno...

Some results about double cyclic codes over 𝔽_q+v𝔽_q+v^2𝔽_q

Let 𝔽_q be the finite field with q elements. This paper mainly researche...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.