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Maximum-distance-separable (MDS) codes and self-dual codes are important families of classical codes in coding theory. It is of interest to investigate MDS self-dual codes. The existence of MDS self-dual codes over finite field F q has been completely solved for q is even. In this paper, for finite field with odd characteristic, we obtain some new results on the existence of MDS self-dual codes by generalized Reed-Solomon (GRS) codes and extended GRS codes. [ABSTRACT FROM AUTHOR]

Any permutation polynomial is an n -cycle permutation. When n is a specific small positive integer, one can obtain efficient permutations, such as involutions, triple-cycle permutations and quadruple-cycle permutations. These permutations have important applications in cryptography and coding theory. Inspired by the AGW Criterion, we propose criteria for n -cycle permutations, which mainly are of the form x r h (x s). We then propose unified constructing methods including recursive ways and a cyclotomic way for n -cycle permutations of such form. We demonstrate our approaches by constructing three classes of explicit triple-cycle permutations with high index and two classes of n -cycle permutations with low index, many of which are new both at levels of permutation property and cycle property. [ABSTRACT FROM AUTHOR]

In this paper, we explicitly determine the generator polynomials of all repeated-root constacyclic codes of length n l p s over F q and their dual codes, where l is an odd prime different from p , and n is an odd prime different from both l and p such that n = 2 h + 1 for some prime h or h = 1 . Moreover, we give all self-dual cyclic codes of length n l p s over F q and their enumeration. All linear complementary dual (LCD) cyclic codes of length n l p s over F q and their enumeration are also obtained. [ABSTRACT FROM AUTHOR]