Repeated-root Negacyclic Codes Over Z_2～a

Classical coding theory takes place in the setting of vector spaces over finite fields. In the 1990s, the theory of error-correcting codes over finite rings has experienced tremendous growth since the significant discovery that several well-known prominent families of good nonlinear binary codes can be identified as images of linear codes over Z₄ under the Gray map. Since then, codes over finite rings have been given more attention. In this paper, we study negacyclic codes of length 2^kn(k≥1,n is odd) over the integers modulo 2^a and their duals, and investigate some properties of self-dual negacyclic codes. We also study the Hamming distances and homogeneous distances of negacyclic codes of 2^k over GR(2^a,m). The details are given as follows:1. Using the discrete Fourier transform, we give the generators of negacyclic codes of length 2^k n over Z_2^a. and the enumeration formula of the number of suchnegacyclic codes.2. Using the Mattson-Solomon polynomial of a codeword, we derive the generators of the duals of negacyclic codes of length 2^kn over Z_2^a.3. Some properties of self-dual negacyclic codes of length 2^k n over Z_2^a are obtained, and a necessary and sufficient condition for the existence of self-dual negacyclic codes of length 2^kn over Z_2^a is proved. We list nontrivial self-dualnegacyclic codes over Z₄ of length≤60, whose Gray images are binary linearself-dual cyclic codes.4. The Hamming distances and homogeneous distances of all negacyclic codesof length 2^k over the Galois ring GR(2^a,m) are given. In particular, the Leedistances of all negacyclic codes over Z₄ of length 2^k are obtained. The Gray images of such negacyclic codes over Z₄ are also determined under the Gray map.