On Some Classes of Repeated-root Constacyclic Codes of Length a Power of 2 over Galois Rings
Hai Quang Dinh · Birkhäuser Basel eBooks · 2010
Negacyclic codes of length 2 s over the Galois ring GR(2 a , m) are ideals of the chain ring $$ \frac{{GR \left( {2^a ,m} \right)\left[ x \right]}} {{\left\langle {x^{2^s } + 1} \right\rangle }} $$ . This structure is used to provide the Hamming and homogeneous distances of all such negacyclic codes. The technique is then generalized to obtain the structure and Hamming and homogeneous distances of all γ-constacyclic codes of length 2 s over GR(2 a , m), where γ is any unit of the ring GR(2 a , m) that has the form γ = (4k 0 −1)+4k 1ξ +...+ 4k m−1ξ m−1, for integers k 0, k 1, ..., k m−1. Among other results, duals of such γ-constacyclic codes are studied, and necessary and sufficient conditions for the existence of a self-dual γ-constacyclic code are established.