共查询到20条相似文献,搜索用时 31 毫秒
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The finite field analog of the classical Kakeya problem asks the smallest possible size for a set of points in the Desarguesian affine plane which contains a line in every direction. This problem has been definitively solved by Blokhuis and Mazzocca (2008), who show that in , a prime power, the smallest possible size of such a set is . In this paper we examine a new construction of an infinite family of sets in containing a line in every direction, where with a prime power and an integer with . These sets have size , which is small in the sense that the lower bound is also plus smaller order terms. In addition, we discuss the minimality of our sets, showing that they contain no proper subset containing a line in every direction. 相似文献
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The first author showed that the list chromatic number of every graph with average degree is at least . We prove that for , every -uniform hypergraph in which at least half of the -vertex subsets are contained in at least edges has list chromatic number at least . When is fixed, this is sharp up to a constant factor. 相似文献
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Yian Xu 《Discrete Mathematics》2017,340(12):2972-2977
Bamberg and Giudici (2011) showed that the point graphs of certain generalised quadrangles of order , where is a prime power with , are both normal and non-normal Cayley graphs for two isomorphic groups. We call these graphs BG-graphs. In this paper, we show that the Cayley graphs obtained from a finite number of BG-graphs by Cartesian product, direct product, and strong product also possess the property of being normal and non-normal Cayley graphs for two isomorphic groups. 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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Ye Wang 《Discrete Mathematics》2017,340(12):2782-2788
Let be the finite field of elements for prime power and let be the character of . For any positive integer , the linearized Wenger graph is defined as follows: it is a bipartite graph with the vertex partitions being two copies of the -dimensional vector space , and two vertices and being adjacent if , for all . In this paper, we show that for any positive integers and with , contains even cycles of length which is an open problem put forward by Cao et al. (2015). 相似文献
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For integers , a -coloring of a graph is a proper coloring with at most colors such that for any vertex with degree , there are at least min different colors present at the neighborhood of . The -hued chromatic number of , , is the least integer such that a -coloring of exists. The list-hued chromatic number of is similarly defined. Thus if , then . We present examples to show that, for any sufficiently large integer , there exist graphs with maximum average degree less than 3 that cannot be -colored. We prove that, for any fraction , there exists an integer such that for each , every graph with maximum average degree is list -colorable. We present examples to show that for some there exist graphs with maximum average degree less than 4 that cannot be -hued colored with less than colors. We prove that, for any sufficiently small real number , there exists an integer such that every graph with maximum average degree satisfies . These results extend former results in Bonamy et al. (2014). 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 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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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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Jakub Przybyło 《Discrete Mathematics》2017,340(10):2402-2407
Consider a positive integer and a graph with maximum degree and without isolated edges. The least so that a proper edge colouring exists such that for every pair of distinct vertices at distance at most in is denoted by . For , it has been proved that . For any in turn an infinite family of graphs is known with . We prove that, on the other hand, for . In particular, we show that if . 相似文献
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Danila Cherkashin 《Discrete Mathematics》2018,341(3):652-657
This paper studies the quantity , that is the minimal number of edges of an -uniform hypergraph without panchromatic coloring (it means that every edge meets every color) in colors. If then all bounds have a type , where , are some algebraic fractions. The main result is a new lower bound on when is at least ; we improve an upper bound on if .Also we show that has upper and lower bounds depending only on when the ratio is small, which cannot be reached by the previous probabilistic machinery.Finally we construct an explicit example of a hypergraph without panchromatic coloring and with edges for . 相似文献
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