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).
Institute of Electrical and Electronics Engineers
Dinh, Hai Q (2009). Constacyclic Codes of Length 2s Over Galois Extension Rings of F2 + uF2. Institute of Electrical and Electronics Engineers 55(4) 1730-1740. doi: 10.1109/TIT.2009.2013015. Retrieved from https://oaks.kent.edu/mathpubs/21