Constacyclic codes over

We study the structure of ( 1 + u ) -constacyclic codes of an arbitrary length n over the ring F 2 + uF 2 . We find a set of generators for each ( 1 + u ) -constacyclic code and its dual. We study the rank of cyclic codes and find their minimal spanning sets. We prove that the Gray image of a ( 1 + u ) -constacyclic code is a binary cyclic code of length 2 n . We conclude by giving examples of constacyclic codes and their Gray image binary codes. We give a direct construction of a [12,7,4] linear binary cyclic code that match the Hamming distance of the best binary code with length 12 and dimension 7.