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《Discrete Mathematics》2022,345(9):112970
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We consider four classes of polynomials over the fields , , , , , , , where . We find sufficient conditions on the pairs for which these polynomials permute and we give lower bounds on the number of such pairs. 相似文献
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Let M be a random rank-r matrix over the binary field , and let be its Hamming weight, that is, the number of nonzero entries of M.We prove that, as with r fixed and tending to a constant, we have that converges in distribution to a standard normal random variable. 相似文献
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Cyclic code is an interesting topic in coding theory and communication systems. In this paper, we investigate the ternary cyclic codes with parameters based on some results proposed by Ding and Helleseth in 2013. Six new classes of optimal ternary cyclic codes are presented by determining the solutions of certain equations over . 相似文献
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In this paper, based on the structure of embedded fields, we investigate explicit construction of systematic mMDS sliding window codes with memory . First, over GF() with and , we propose an algorithm to construct mMDS codes with memory 2, which are optimal in the sense that is the maximum possible value of k for a sliding window code with memory 2 over GF() to be mMDS. When , every constructed code has the extra property that it contains a mMDS sliding window code with memory 2 as a subcode over the subfield GF(). Next, over GF() with and , we introduce a method to construct mMDS codes memory 3, and a few new codes have been obtained consequently. When , every code constructed by the new approach also has the property that it contains an mMDS subcode over the subfield GF(). The embedding subfield-subcode property enhances the flexibility and efficiency of the designed codes. 相似文献
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After a brief review of the existing results on permutation binomials of finite fields, we introduce the notion of equivalence among permutation binomials (PBs) and describe how to bring a PB to its canonical form under equivalence. We then focus on PBs of of the form , where n and d are positive integers and . Our contributions include two nonexistence results: (1) If q is even and sufficiently large and , then is not a PB of . (2) If , q is sufficiently large and , then is not a PB of under certain additional conditions. (1) partially confirms a recent conjecture by Tu et al. (2) is an extension of a previous result with . 相似文献
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In this paper, we give the dimension and the minimum distance of two subclasses of narrow-sense primitive BCH codes over with designed distance for all , where q is a prime power and is a positive integer. As a consequence, we obtain an affirmative answer to two conjectures proposed by C. Ding in 2015. Furthermore, using the previous part, we extend some results of Yue and Hu [16], and we give the dimension and, in some cases, the Bose distance for a large designed distance in the range for , where if m is odd, and if m is even. 相似文献
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