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A Butson-type Hadamard matrix is an matrix over the complex th roots of unity that fulfils . It is well known that a matrix can be used to construct a matrix, that is, a real Hadamard matrix. This method is here generalised to construct a matrix from a matrix, where has at most two distinct prime divisors, one of them being . Moreover, an algorithm for finding the domain of the mapping from its codomain in the case is developed and used to classify the matrices from a classification of the matrices. 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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Let q be a positive integer. Recently, Niu and Liu proved that, if , then the product is not a powerful number. In this note, we prove (1) that, for any odd prime power ? and , the product is not a powerful number, and (2) that, for any positive odd integer ?, there exists an integer such that, for any positive integer , the product is not a powerful number. 相似文献
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Sunghyu Han Jon-Lark Kim Heisook Lee Yoonjin Lee 《Finite Fields and Their Applications》2012,18(3):613-633
There is a one-to-one correspondence between ?-quasi-cyclic codes over a finite field and linear codes over a ring . Using this correspondence, we prove that every ?-quasi-cyclic self-dual code of length m? over a finite field can be obtained by the building-up construction, provided that or , m is a prime p, and q is a primitive element of . We determine possible weight enumerators of a binary ?-quasi-cyclic self-dual code of length p? (with p a prime) in terms of divisibility by p. We improve the result of Bonnecaze et al. (2003) [3] by constructing new binary cubic (i.e., ?-quasi-cyclic codes of length 3?) optimal self-dual codes of lengths (Type I), 54 and 66. We also find quasi-cyclic optimal self-dual codes of lengths 40, 50, and 60. When , we obtain a new 8-quasi-cyclic self-dual code over and a new 6-quasi-cyclic self-dual code over . When , we find a new 4-quasi-cyclic self-dual code over and a new 6-quasi-cyclic self-dual code over . 相似文献
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Let e be a positive integer, p be an odd prime, , and be the finite field of q elements. Let . The graph is a bipartite graph with vertex partitions and , and edges defined as follows: a vertex is adjacent to a vertex if and only if and . If and , the graph contains no cycles of length less than eight and is edge-transitive. Motivated by certain questions in extremal graph theory and finite geometry, people search for examples of graphs containing no cycles of length less than eight and not isomorphic to the graph , even without requiring them to be edge-transitive. So far, no such graphs have been found. It was conjectured that if both f and g are monomials, then no such graphs exist. In this paper we prove the conjecture. 相似文献
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Jianbei An Heiko Dietrich Shih-Chang Huang 《Journal of Pure and Applied Algebra》2018,222(12):4020-4039
We consider the finite exceptional group of Lie type (universal version) with , where and . We classify, up to conjugacy, all maximal-proper 3-local subgroups of G, that is, all 3-local which are maximal with respect to inclusion among all proper subgroups of G which are 3-local. To this end, we also determine, up to conjugacy, all elementary-abelian 3-subgroups containing , all extraspecial subgroups containing , and all cyclic groups of order 9 containing . These classifications are an important first step towards a classification of the 3-radical subgroups of G, which play a crucial role in many open conjectures in modular representation theory. 相似文献
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A vertex-deleted subgraph of a graph G is a card. A dacard specifies the degree of the deleted vertex along with the card. The adversary degree-associated reconstruction number is the least k such that every set of k dacards determines G. We determine , where the double-broom with is the tree with vertices obtained from a path with p vertices by appending m leaves at one end and n leaves at the other end. We determine for all . For , usually , except and . There are exceptions when or . For the usual value is 4, with exceptions when or . 相似文献
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In this paper, we show that for any fixed integers and , the star-critical Ramsey number for all sufficiently large . Furthermore, for any fixed integers and , as . 相似文献
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Gábor Korchmáros Maria Montanucci Pietro Speziali 《Journal of Pure and Applied Algebra》2018,222(7):1810-1826
Let be the algebraic closure of a finite field of odd characteristic p. For a positive integer m prime to p, let be the transcendence degree 1 function field defined by . Let and . The extension is a non-Galois extension. Let K be the Galois closure of F with respect to H. By Stichtenoth [20], K has genus , p-rank (Hasse–Witt invariant) and a -automorphism group of order at least . In this paper we prove that this subgroup is the full -automorphism group of K; more precisely where Δ is an elementary abelian p-group of order and D has an index 2 cyclic subgroup of order . In particular, , and if K is ordinary (i.e. ) then . On the other hand, if G is a solvable subgroup of the -automorphism group of an ordinary, transcendence degree 1 function field L of genus defined over , then ; see [15]. This shows that K hits this bound up to the constant .Since has several subgroups, the fixed subfield of such a subgroup N may happen to have many automorphisms provided that the normalizer of N in is large enough. This possibility is worked out for subgroups of Δ. 相似文献