Journal•ISSN: 0097-3165

# Journal of Combinatorial Theory, Series A

About: Journal of Combinatorial Theory, Series A is an academic journal. The journal publishes majorly in the area(s): Conjecture & Upper and lower bounds. It has an ISSN identifier of 0097-3165. Over the lifetime, 5386 publication(s) have been published receiving 122443 citation(s).

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TL;DR: The polynomial degree of a bent function P ( x ) is studied, as are the properties of the Fourier transform of (−1) P(x) , and a connection with Hadamard matrices.

Abstract: Let P ( x ) be a function from GF (2 n ) to GF (2). P ( x ) is called “bent” if all Fourier coefficients of (−1) P(x) are ±1. The polynomial degree of a bent function P ( x ) is studied, as are the properties of the Fourier transform of (−1) P(x) , and a connection with Hadamard matrices.

1,134 citations

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TL;DR: Asymptotics are obtained for the number of n × n symmetric non-negative integer matrices subject to the following constraints: each row sum is specified and bounded, and a specified “sparse” set of entries must be zero.

Abstract: Asymptotics are obtained for the number of n × n symmetric non-negative integer matrices subject to the following constraints: (i) each row sum is specified and bounded, (ii) the entries are bounded, and (iii) a specified “sparse” set of entries must be zero. The result can be interpreted in terms of incidence matrices for labeled graphs.

973 citations

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TL;DR: This paper will answer the question in the affirmative by determining the exact upper bound of T if T is a family of subsets of some infinite set S then either there exists to each number n a set A ⊂ S with |A| = n such that |T ∩ A| = 2n or there exists some number N such that •A| c for each A⩾ N and some constant c.

Abstract: If T is a family of sets and A some set we denote by T ∩ A the following family of subsets of A: T ∩ A = {F ∩ A; F ϵ T}. P. Erdos (oral communication) transmitted to me in Nice the following question: Is it true that if T is a family of subsets of some infinite set S then either there exists to each number n a set A ⊂ S with |A| = n such that |T ∩ A| = 2n or there exists some number N such that |T ∩ A| ⩽ |A|c for each A ⊂ S with |A| ⩾ N and some constant c? In this paper we will answer this question in the affirmative by determining the exact upper bound. (Theorem 2).1

937 citations

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TL;DR: If the simplicial complex formed by the neighborhoods of points of a graph is (k − 2)-connected then the graph is not k-colorable, and Kneser's conjecture is proved, asserting that if all n-subsets of a (2n − k)-element set are divided into k + 1 classes, one of the classes contains two disjoint n- subsets.

Abstract: If the simplicial complex formed by the neighborhoods of points of a graph is ( k − 2)-connected then the graph is not k -colorable. As a corollary Kneser's conjecture is proved, asserting that if all n -subsets of a (2n − k) -element set are divided into k + 1 classes, one of the classes contains two disjoint n -subsets.

827 citations

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TL;DR: The characters of the adjacency algebra of Ω, which yield the MacWilliams transform on q-distance enumerators, are expressed in terms of generalized Krawtchouk polynomials.

Abstract: Let Ω be the set of bilinear forms on a pair of finite-dimensional vector spaces over GF(q). If two bilinear forms are associated according to their q-distance (i.e., the rank of their difference), then Ω becomes an association scheme. The characters of the adjacency algebra of Ω, which yield the MacWilliams transform on q-distance enumerators, are expressed in terms of generalized Krawtchouk polynomials. The main emphasis is put on subsets of Ω and their q-distance structure. Certain q-ary codes are attached to a given X ⊂ Ω ; the Hamming distance enumerators of these codes depend only on the q-distance enumerator of X. Interesting examples are provided by Singleton systems X ⊂ Ω , which are defined as t-designs of index 1 in a suitable semilattice (for a given integer t). The q-distance enumerator of a Singleton system is explicitly determined from the parameters. Finally, a construction of Singleton systems is given for all values of the parameters.

643 citations