81 research outputs found

    Zero-error capacity of binary channels with memory

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    We begin a systematic study of the problem of the zero--error capacity of noisy binary channels with memory and solve some of the non--trivial cases.Comment: 10 pages. This paper is the revised version of our previous paper having the same title, published on ArXiV on February 3, 2014. We complete Theorem 2 of the previous version by showing here that our previous construction is asymptotically optimal. This proves that the isometric triangles yield different capacities. The new manuscript differs from the old one by the addition of one more pag

    A group-theoretic approach to fast matrix multiplication

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    We develop a new, group-theoretic approach to bounding the exponent of matrix multiplication. There are two components to this approach: (1) identifying groups G that admit a certain type of embedding of matrix multiplication into the group algebra C[G], and (2) controlling the dimensions of the irreducible representations of such groups. We present machinery and examples to support (1), including a proof that certain families of groups of order n^(2 + o(1)) support n-by-n matrix multiplication, a necessary condition for the approach to yield exponent 2. Although we cannot yet completely achieve both (1) and (2), we hope that it may be possible, and we suggest potential routes to that result using the constructions in this paper.Comment: 12 pages, 1 figure, only updates from previous version are page numbers and copyright informatio

    Strong Qualitative Independence

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    AbstractThe subsets A,B of the n-element X are said to be s-strongly separating if the two sets divide X into four sets of size at least s. The maximum number h(n,s) of pairwise s-strongly separating subsets was asymptotically determined by Frankl (Ars Combin. 1 (1976) 53) for fixed s and large n. A new proof is given. Also, estimates for h(n,cn) are found where c is a small constant

    Local chromatic number and Sperner capacity

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    We introduce a directed analog of the local chromatic number defined by Erdos et al. [Discrete Math. 59 (1986) 21-34] and show that it provides an upper bound for the Sperner capacity of a directed graph. Applications and variants of this result are presented. In particular, we find a special orientation of an odd cycle and show that it achieves the maximum of Sperner capacity among the differently oriented versions of the cycle. We show that apart from this orientation, for all the others an odd cycle has the same Sperner capacity as a single edge graph. We also show that the (undirected) local chromatic number is bounded from below by the fractional chromatic number while for power graphs the two invariants have the same exponential asymptotics (under the co-normal product on which the definition of Sperner capacity is based). We strengthen our bound on Sperner capacity by introducing a fractional relaxation of our directed variant of the local chromatic number. (C) 2005 Elsevier Inc. All rights reserved

    Dilworth rate: a generalization of Witsenhausen's zero-error rate for directed graphs

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    Combinatorics

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    A Survey of Binary Covering Arrays

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    Binary covering arrays of strength t are 0–1 matrices having the property that for each t columns and each of the possible 2[superscript t] sequences of t 0's and 1's, there exists a row having that sequence in that set of t columns. Covering arrays are an important tool in certain applications, for example, in software testing. In these applications, the number of columns of the matrix is dictated by the application, and it is desirable to have a covering array with a small number of rows. Here we survey some of what is known about the existence of binary covering arrays and methods of producing them, including both explicit constructions and search techniques
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