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In this work, we propose two criteria for linear codes obtained from the Plotkin sum construction being symplectic self-orthogonal (SO) and linear complementary dual (LCD). As specific constructions, several classes of symplectic SO codes with good parameters including symplectic maximum distance separable codes are derived via ℓ -intersection pairs of linear codes and generalized Reed-Muller codes. Also symplectic LCD codes are constructed from general linear codes. Furthermore, we obtain some binary symplectic LCD codes, which are equivalent to quaternary trace Hermitian additive complementary dual codes that outperform the best-known quaternary Hermitian LCD codes reported in the literature. In addition, we prove that symplectic SO and LCD codes obtained in these ways are asymptotically good. [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]

Let q = p h be a prime power and e be an integer with 0 ≤ e ≤ h − 1. e -Galois self-orthogonal codes are generalizations of Euclidean self-orthogonal codes (e = 0) and Hermitian self-orthogonal codes (e = h 2 and h is even). In this paper, we propose two general methods to construct e -Galois self-orthogonal (extended) generalized Reed-Solomon (GRS) codes. As a consequence, eight new classes of e -Galois self-orthogonal (extended) GRS codes with odd q and 2 e | h are obtained. Based on the Galois dual of a code, we also study its punctured and shortened codes. As applications, new e ′ -Galois self-orthogonal maximum distance separable (MDS) codes for all possible e ′ satisfying 0 ≤ e ′ ≤ h − 1 , new e -Galois self-orthogonal MDS codes via the shortened codes, and new MDS codes with prescribed dimensional e -Galois hull via the punctured codes are derived. Moreover, some new q -ary quantum MDS codes with length greater than q + 1 and minimum distance greater than q 2 + 1 are obtained. [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]