Title

Constacyclic Codes of Length 2s Over Galois Extension Rings of F2 + uF2

Publication Title

IEEE Transactions on Information Theory

Publication Date

4-2009

Document Type

Article

DOI

10.1109/TIT.2009.2013015

Keywords

repeated-root codes, Codes over rings, constacyclic codes, cyclic codes, Hamming distance, Galois extension, mass formula, Galois fields, Concatenated codes, Linear code, Algebra

Disciplines

Algebra | Mathematics

Abstract

We study all constacyclic codes of length 2s over GR(Rfr,m), the Galois extension ring of dimension m of the ring Rfr=F2+uF2. The units of the ring GR(Rfr,m) are of the forms alpha, and alpha+ubeta, where alpha, beta are nonzero elements of F2m, which correspond to 2 m(2m-1) such constacyclic codes. First, the structure and Hamming distances of (1+ugamma)-constacyclic codes are established. We then classify all cyclic codes of length 2sover GR(Rfr,m), and obtain a formula for the number of those cyclic codes, as well as the number of codewords in each code. Finally, one-to-one correspondences between cyclic and alpha-constacyclic codes, as well as (1+ugamma)-constacyclic and (alpha+ubeta) -constacyclic codes are provided via ring isomorphisms, that allow us to carry over the results about cyclic and (1+ugamma)-constacyclic accordingly to all constacyclic codes of length 2s over GR(Rfr,m).

Publisher

Institute of Electrical and Electronics Engineers

Publisher Location

New York, NY


Share

COinS