Constacyclic Codes of Length $2^s$ Over Galois Extension Rings of ${\BBF}_{2}+u{\BBF}_2$

Abstract

We study all constacyclic codes of length 2^s 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(2^m-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 2^s over 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 2^s over GR(Rfr,m).