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Recently, the research of trace codes over finite fields has received a lot of attention because some of these codes have good parameters. In this paper, we construct several classes of trace codes from some special classes of linear codes and give their weight distributions, showing that they contain some optimal and almost optimal codes. Furthermore, we determine the parameters of the duals of the trace codes, and show that they are distance-optimal or nearly distance-optimal with respect to the sphere-packing bound. [ABSTRACT FROM AUTHOR]

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]