Researches On The Duality Of Constacyclic Codes

In coding theory, cyclic codes are an extremely important class of codes which have been extensively studied. Constacyclic codes are a natural generalization of cyclic codes, and inherit most of the advantages of cyclic codes. The duality plays an important role in coding theory, which has important applications in the researches of weight structure and algebraic structure. This dessertation is focusing on three topics about dualities of constacyclic codes(including cyclic codes).1.Generalizing Euclidean inner product and Hermitian inner product, we intro-duce Galois inner product in term of any automorphism of finite fields. We study Galois self-duality of constacyclic codes in a very general setting (including cyclic case, nonsemisimple case) by a uniform method. The uniform method includes:we define q-coset functions to characterize constacyclic codes, which generalize charac-terization of constacyclic codes by sets of zeros at the semisimple case; we define new isometries between constacyclic codes which can be applied in any case including the nonsimisimple case. In particular, we obtain a necessary and sufficient condition for the existence of Galois self-dual and isometrically Galois self-dual constacyclic codes. As consequences, the related results on self-dual, iso-dual and Hermitian self-dual constacyclic codes can be derived.2.Generalizing even-like duadic cyclic codes and Type-Ⅱ duadic negacyclic codes, we introduce even-like (i.e., Type-Ⅱ) and odd-like duadic constacyclic codes, and study their properties and existence. We show that the duals of even-like duadic constacyclic codes are odd-like duadic constacyclic codes and vice versa. We exhibit necessary and sufficient conditions for the existence of even-like duadic constacyclic codes. Besides, we proof that if Type-I duadic λ-constacyclic codes of length n do not exist but Type-Ⅱ duadic λ-constacyclic codes of length n exist, then any even-like pair of Type-Ⅱ duadic A-constacyclic codes of length n is a maximal iso-orthogonal pair of λ-constacyclic codes. Finally, we show that some alternant MDS-codes can be constructed from Type-Ⅱ duadic constacyclic codes.3.We introduce a new class of isometries to study isometrically self-dual cyclic codes. In the existing literature, almost all isometries of cyclic codes are constructed by the permutations of Zn (the residue ring of integers modulo n) induced by mul-tiplication of Zn, called multipliers. The isometries induced by multipliers can’t be used to study isometrically self-dual cyclic codes. We define a new class of isometries between cyclic codes by the permutations of Zn induced by addition of Zn, called translation operators. Using new translation operators, we give necessary and suf-ficient conditions for the existence of isometrically self-dual cyclic and constacyclic codes, and study their properties.