Constacyclic Codes of Length $2^s$ Over Galois Extension Rings of ${\BBF}_{2}+u{\BBF}_2$
Hai Quang Dinh · IEEE Transactions on Information Theory · 2009
We study all constacyclic codes of length 2sover GR(Rfr,m), the Galois extension ring of dimension m of the ringRfr=F2+uF2. The units of the ring GR(Rfr,m) are of the formsalpha, andalpha+ubeta, wherealpha,betaare nonzero elements of F2m, which correspond to2m(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 2soverGR(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 andalpha-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 2soverGR(Rfr,m).