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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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《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 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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In this work we give a characterization of Galois Linear Complementary Dual codes and Galois-invariant codes over mixed alphabets of finite chain rings, which leads to the study of the Gray image of -linear codes, where and that provides LCD codes over . 相似文献
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We introduce an altered version of the four circulant construction over group rings for self-dual codes. We consider this construction over the binary field, the rings and ; using groups of order 4 and 8. Through these constructions and their extensions, we find binary self-dual codes of lengths 16, 32, 48, 64 and 68, many of which are extremal. In particular, we find forty new extremal binary self-dual codes of length 68, including twelve new codes with in , which is the first instance of such a value in the literature. 相似文献
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An important problem on almost perfect nonlinear (APN) functions is the existence of APN permutations on even-degree extensions of larger than 6. Browning et al. (2010) gave the first known example of an APN permutation on the degree-6 extension of . The APN permutation is CCZ-equivalent to the previously known quadratic Kim κ-function (Browning et al. (2009)). Aside from the computer based CCZ-inequivalence results on known APN functions on even-degree extensions of with extension degrees less than 12, no theoretical CCZ-inequivalence result on infinite families is known. In this paper, we show that Gold and Kasami APN functions are not CCZ-equivalent to permutations on infinitely many even-degree extensions of . In the Gold case, we show that Gold APN functions are not equivalent to permutations on any even-degree extension of , whereas in the Kasami case we are able to prove inequivalence results for every doubly-even-degree extension of . 相似文献
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《Journal of Pure and Applied Algebra》2019,223(11):4583-4591
Marks showed that , the group algebra over the quaternion group, is a reversible nonsymmetric ring, then questioned whether or not this ring is minimal with respect to cardinality. In this work, it is shown that the cardinality of a minimal reversible nonsymmetric ring is indeed 256. Furthermore, it is shown that although is a duo ring, there are also examples of minimal reversible nonsymmetric rings which are nonduo. 相似文献
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《Discrete Mathematics》2019,342(3):697-714
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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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We show that the construction of Gabor frames in with generators in and with respect to time-frequency shifts from a rectangular lattice is equivalent to the construction of certain Gabor frames for over the adeles over the rationals and the group . Furthermore, we detail the connection between the construction of Gabor frames on the adeles and on with the construction of certain Heisenberg modules. 相似文献
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Julia Semikina 《Journal of Pure and Applied Algebra》2019,223(10):4509-4523
I. Hambleton, L. Taylor and B. Williams conjectured a general formula in the spirit of H. Lenstra for the decomposition of for any finite group G and noetherian ring R. The conjectured decomposition was shown to hold for some large classes of finite groups. D. Webb and D. Yao discovered that the conjecture failed for the symmetric group , but remarked that it still might be reasonable to expect the HTW-decomposition for solvable groups. In this paper we show that the solvable group is also a counterexample to the conjectured HTW-decomposition. Nevertheless, we prove that for any finite group G the rank of does not exceed the rank of the expression in the HTW-decomposition. We also show that the HTW-decomposition predicts correct torsion for for any finite group G. Furthermore, we prove that for any degree other than the conjecture gives a correct prediction for the rank of . 相似文献