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In this paper, a set of generators (in a unique from) called the distinguished set of generators, of a cyclic code C of length n = 2 k (where k is a natural number) over Z 2 m is obtained. This set of generators is used to find the rank of the cyclic code C. It is proved that the rank of a cyclic code C of length n = 2 k over Z 2 m is equal to n − v , where v is the degree of a minimal degree polynomial in C. Then a description of all MHDR (maximum hamming distance with respect to rank) cyclic codes of length n = 2 k over Z 2 m is given. An example of the best cyclic codes over Z 8 of length 4 having largest minimum Hamming, Lee and Euclidean distances among all cyclic codes of the same rank is also given. Further, using an isomorphism between cyclic and negacyclic codes of odd length over finite chain rings given by Dinh, Lopez and Szabo, the above results are extended to cyclic and negacyclic codes of length p k over Z p m for an odd prime p. [ABSTRACT FROM AUTHOR]

Due to the wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an interesting research topic in coding theory. In this paper, let p be a prime with p ≥ 7. We determine the Hamming distances of all repeated-root cyclic codes of length 5 p s over F q , where q = p m and both s and m are positive integers. Furthermore, we find all MDS cyclic codes of length 5 p s and take quantum synchronizable codes from repeated-root cyclic codes of length 5 p s . By comparing the minimum distances of 5 p s -length repeated-root cyclic codes to BCH codes of close lengths, we illustrated that quantum synchronizable codes constructed from repeated-root cyclic codes have in general better performance in correcting Pauli errors than those from BCH codes. [ABSTRACT FROM AUTHOR]