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1.
Let be a finite field of odd order and , where are positive integers, are distinct odd primes and . In this paper, we study the irreducible factorization of over and all primitive idempotents in the ring .Moreover, we obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length over . 相似文献
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Let be the Galois ring of characteristic and cardinality . Firstly, we give all primitive idempotent generators of irreducible cyclic codes of length over , and a -adic integer ring with . Secondly, we obtain all primitive idempotents of all irreducible cyclic codes of length over , where and are three primes with , , and . Finally, as applications, weight distributions of all irreducible cyclic codes for and generator polynomials of self-dual cyclic codes of length and over are given. 相似文献
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Anuradha Sharma Gurmeet K. Bakshi Madhu Raka 《Finite Fields and Their Applications》2007,13(4):1086-1095
Let m be a positive integer and q be an odd prime power. In this paper, the weight distributions of all the irreducible cyclic codes of length 2m over Fq are determined explicitly. 相似文献
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After several remarks on two-weight irreducible cyclic codes, we introduce a family of projective two-weight cyclic codes and a family of projective two-weight constacyclic codes and we discuss the existence of such codes. 相似文献
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The objectives of this paper are to survey and extend earlier results on the weight distributions of irreducible cyclic codes, present a divisibility theorem and develop bounds on the weights in irreducible cyclic codes. 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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We introduce new classes of 2-weight cyclic codes which are direct sums of 1-weight irreducible cyclic codes
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Hai Q. Dinh Xiaoqiang Wang Hongwei Liu Songsak Sriboonchitta 《Discrete Mathematics》2019,342(5):1456-1470
Let be an odd prime, , be positive integers, be nonzero elements of the finite field such that . In this paper, we show that, for any positive integer , the Hamming distances of all repeated-root -constacyclic codes of length can be determined by those of certain simple-root -constacyclic codes of length . Using this result, Hamming distances of all constacyclic codes of length are obtained. As an application, we identify all MDS -constacyclic codes of length . 相似文献
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It is well known that the Stickelberger–Swan theorem is very important for determining the reducibility of polynomials over a binary field. Using this theorem the parity of the number of irreducible factors for some kinds of polynomials over a binary field, for instance, trinomials, tetranomials, self-reciprocal polynomials and so on was determined. We discuss this problem for Type II pentanomials, namely xm+xn+2+xn+1+xn+1F2[x] for even m. Such pentanomials can be used for the efficient implementation of multiplication in finite fields of characteristic two. Based on the computation of the discriminant of these pentanomials with integer coefficients, we will characterize the parity of the number of irreducible factors over F2 and establish necessary conditions for the existence of this kind of irreducible pentanomials.Our results have been obtained in an experimental way by computing a significant number of values with Mathematica and extracting the relevant properties. 相似文献
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Ruihu Li 《Discrete Mathematics》2008,308(9):1603-1611
In this paper, we construct a large number of good quantum codes of minimum distances five and six by Steane's Construction. Our methods involve the study of the check matrices of binary extended BCH-codes, together with puncturing and combining such matrices. 相似文献
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Explicit expressions for all the 3n+2 primitive idempotents in the ring Rpnq=GF(ℓ)[x]/(xpnq−1), where p,q,ℓ are distinct odd primes, ℓ is a primitive root modulo pn and q both,
, are obtained. The dimension, generating polynomials and the minimum distance of the minimal cyclic codes of length pnq over GF(ℓ) are also discussed. 相似文献
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An index , length quasi-cyclic code can be viewed as a cyclic code of length over the field via a basis of the extension . However, this cyclic code is only linear over , making it an additive cyclic code, or an -linear cyclic code, over the alphabet . This approach was recently used in Shi et al. (2017) [16] to study a class of quasi-cyclic codes, and more importantly in Shi et al. (2017) [17] to settle a long-standing question on the asymptotic performance of cyclic codes. Here, we answer one of the problems posed in these two articles, and characterize those quasi-cyclic codes which have -linear cyclic images under a basis of the extension . Our characterizations are based on the module structure of quasi-cyclic codes, as well as on their CRT decompositions into constituents. In the case of a polynomial basis, we characterize the constituents by using the theory of invariant subspaces of operators. We also observe that analogous results extend to the case of quasi-twisted codes. 相似文献
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Let be the finite field with q elements, and T a positive integer. In this article, we find an asymptotic formula for the total number of monic irreducible binomials in of degree less or equal to T, when T is large enough. We also show explicit lower and upper bounds for the number of binomials in the case when T is small. 相似文献