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《Discrete Mathematics》2022,345(11):113059
Let be the finite field of q elements and let be the dihedral group of 2n elements. Left ideals of the group algebra are known as left dihedral codes over of length 2n, and abbreviated as left -codes. Let . In this paper, we give an explicit representation for the Euclidean hull of every left -code over . On this basis, we determine all distinct Euclidean LCD codes and Euclidean self-orthogonal codes which are left -codes over . In particular, we provide an explicit representation and a precise enumeration for these two subclasses of left -codes and self-dual left -codes, respectively. Moreover, we give a direct and simple method for determining the encoder (generator matrix) of any left -code over , and present several numerical examples to illustrative our applications. 相似文献
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The -additive codes are subgroups of , and can be seen as linear codes over when , -additive codes when , or -additive codes when . A -linear generalized Hadamard (GH) code is a GH code over which is the Gray map image of a -additive code. Recursive constructions of -additive GH codes of type with are known. In this paper, we generalize some known results for -linear GH codes with to any prime when , and then we compare them with the ones obtained when . First, we show for which types the corresponding -linear GH codes are nonlinear over . Then, for these codes, we compute the kernel and its dimension, which allow us to classify them completely. Moreover, by computing the rank of some of these codes, we show that, unlike -linear Hadamard codes, the -linear GH codes are not included in the family of -linear GH codes with when prime. Indeed, there are some families with infinite nonlinear -linear GH codes, where the codes are not equivalent to any -linear GH code with . 相似文献
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Let be a prime. We show that, for each integer d with , there exists a generalized almost perfect nonlinear (GAPN) binomial or trinomial over of algebraic degree d. We start by deriving sufficient conditions for the function to be GAPN in the case where one of the terms of G is GAPN. We then give explicit constructions of GAPN binomials over of any odd algebraic degree between p and and, in the case where p is not a Mersenne prime, also of any even algebraic degree in this range. To obtain GAPN functions of even algebraic degree also in the general case, we finally show how to construct GAPN trinomials over of any even algebraic degree between p and by applying a characterization of a special form of GAPN binomials by Özbudak and Sălăgean. Our constructed functions are the first GAPN functions of even algebraic degree over extension fields of odd characteristic reported so far. 相似文献
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We explicitly determine generators of cyclic codes over a non-Galois finite chain ring of length , where p is a prime number and k is a positive integer. We completely classify that there are three types of principal ideals of and four types of non-principal ideals of , which are associated with cyclic codes over of length . We then obtain a mass formula for cyclic codes over of length . 相似文献
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Let be a finite field of cardinality , which is a finite chain ring, and is an odd positive integer. For any , an explicit representation for the dual code of any -constacyclic code over of length is given. And some dual codes of -constacyclic codes over of length 14 are constructed. For the case of , all distinct self-dual -constacyclic codes over of length are determined. 相似文献
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Duc Quang Si 《Journal of Mathematical Analysis and Applications》2021,493(2):124542
Let M be a complete Kähler manifold, whose universal covering is biholomorphic to a ball in (). Our purpose of this article is to establish a non-integrated defect relation with truncated level for a meromorphic mapping on M intersecting a family of hyperplanes in which is non-subdegenerate with respect to the mapping. 相似文献
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We give methods for constructing many self-dual -codes and Type II -codes of length 2n starting from a given self-dual -code and Type II -code of length 2n, respectively. As an application, we construct extremal Type II -codes of length 24 for and extremal Type II -codes of length 32 for . We also construct new extremal Type II -codes of lengths 56 and 64. 相似文献
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《Discrete Mathematics》2020,343(3):111721
The -additive codes are subgroups of , and can be seen as a generalization of linear codes over and . A -linear Hadamard code is a binary Hadamard code which is the Gray map image of a -additive code. A partial classification of these codes by using the dimension of the kernel is known. In this paper, we establish that some -linear Hadamard codes of length are equivalent, once is fixed. This allows us to improve the known upper bounds for the number of such nonequivalent codes. Moreover, up to , this new upper bound coincides with a known lower bound (based on the rank and dimension of the kernel). Finally, when we focus on , the full classification of the -linear Hadamard codes of length is established by giving the exact number of such codes. 相似文献
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We present several existence and nonexistence results for permutation binomials of the form , where and . As a consequence, we obtain a complete characterization of such permutation binomials over , , , , and , where p is an odd prime. 相似文献
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We construct a class of -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 is greater than and the relative distance of the code is convergent to δ, while the rate is convergent to for and . As a consequence, we prove that there exist numerous asymptotically good -additive cyclic codes. 相似文献
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In this paper, we will study Ciani curves in characteristic , in particular their standard forms . It is well-known that any Ciani curve is a non-hyperelliptic curve of genus 3, and its Jacobian variety is isogenous to the product of three elliptic curves. As a main result, we will show that if C is superspecial, then belong to and C is maximal or minimal over . Moreover, in this case we will provide a simple criterion in terms of that tells whether C is maximal (resp. minimal) over . 相似文献
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Let be the finite field of characteristic p with q elements and its extension of degree n. We prove that there exists a primitive element of that produces a completely normal basis of over , provided that with and . 相似文献