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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]

In this paper, we study cyclic self-orthogonal codes over Z 2 m for any odd length. We give the generator polynomials of cyclic self-orthogonal codes over Z 2 m . By using these generator polynomials, we obtain a sufficient and necessary condition for the existence of nontrivial cyclic self-orthogonal codes over Z 2 m and determine the number of such codes for a given odd length. Some results about cyclic self-dual codes over Z 2 m are included. [ABSTRACT FROM AUTHOR]

Let q be a power of an odd prime, and be the trace mapping. A mapping is called planar (or perfect nonlinear) on if for any non-zero , the difference mapping is a permutation where for , . Kyureghyan and Özbudak (2012) [8] considered the planarity of mappings on for and proved that there is no planar for . For the case and , they raised three conjectures. In this paper we prove the third conjecture which says that there is no planar for , by using Kloosterman sums. Our proof also works for case , so we present a new proof of the Kyureghyan–Özbudak result. For case , we present an elementary proof of the first conjecture which says that there is no planar for . [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we explore constacyclic self-dual codes over . We first characterize constacyclic self-dual codes over of any length. Then we determine the structure of η-constacyclic self-dual codes over , where or . This structure is used to find some constacyclic self-dual codes over . [Copyright &y& Elsevier]

Abstract: We study -constacyclic codes over of an arbitrary length, where λ is a unit of and is a positive integer. We first derive the structure of -constacyclic codes of length over and determine the Hamming and homogeneous distances of such constacyclic codes. These codes are then used to classify all -constacyclic codes over of length (n prime to p). In particular, the Gray images of -constacyclic codes over are also discussed. [Copyright &y& Elsevier]

We construct a class of Z p r Z p s -additive cyclic codes generated by pairs of polynomials, where p is a prime number. Based on probabilistic arguments, we determine the asymptotic rates and relative distances of this class of codes: the asymptotic Gilbert-Varshamov bound at 1 + p s − r 2 δ is greater than 1 2 and the relative distance of the code is convergent to δ , while the rate is convergent to 1 1 + p s − r for 0 < δ < 1 1 + p s − r and 1 ≤ r < s. As a consequence, we prove that there exist numerous asymptotically good Z p r Z p s -additive cyclic 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]

Abstract We investigate the minimum distance of q -ary negacyclic codes of length q m − 1 2 generated by a product of two distinct minimal polynomials. A necessary and sufficient condition on the minimum distance of such negacyclic codes is given. Several classes of optimal quinary negacyclic codes with parameters [ (5 m − 1) / 2 , (5 m − 1) / 2 − 2 m , 4 ] are constructed. The dual codes of a subclass of the quinary negacyclic codes are studied. A comparison with cyclic codes is also presented. [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]

Let p ≠ 3 be any prime and l ≠ 3 be any odd prime with gcd ⁡ ( p , l ) = 1 . The multiplicative group F q ⁎ = 〈 ξ 〉 can be decomposed into mutually disjoint union of gcd ⁡ ( q − 1 , 3 l p s ) cosets over the subgroup 〈 ξ 3 l p s 〉 , where ξ is a primitive ( q − 1 ) th root of unity. We classify all repeated-root constacyclic codes of length 3 l p s over the finite field F q into some equivalence classes by this decomposition, where q = p m , s and m are positive integers. According to these equivalence classes, we explicitly determine the generator polynomials of all repeated-root constacyclic codes of length 3 l p s over F q and their dual codes. Self-dual cyclic codes of length 3 l p s over F q exist only when p = 2 . We give all self-dual cyclic codes of length 3 ⋅ 2 s l over F 2 m and their enumeration. We also determine the minimum Hamming distance of these codes when gcd ⁡ ( 3 , q − 1 ) = 1 and l = 1 . [ABSTRACT FROM AUTHOR]