133 research outputs found

    Frequency permutation arrays

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    Motivated by recent interest in permutation arrays, we introduce and investigate the more general concept of frequency permutation arrays (FPAs). An FPA of length n=m lambda and distance d is a set T of multipermutations on a multiset of m symbols, each repeated with frequency lambda, such that the Hamming distance between any distinct x,y in T is at least d. Such arrays have potential applications in powerline communication. In this paper, we establish basic properties of FPAs, and provide direct constructions for FPAs using a range of combinatorial objects, including polynomials over finite fields, combinatorial designs, and codes. We also provide recursive constructions, and give bounds for the maximum size of such arrays.Comment: To appear in Journal of Combinatorial Design

    Group Divisible Codes and Their Application in the Construction of Optimal Constant-Composition Codes of Weight Three

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    The concept of group divisible codes, a generalization of group divisible designs with constant block size, is introduced in this paper. This new class of codes is shown to be useful in recursive constructions for constant-weight and constant-composition codes. Large classes of group divisible codes are constructed which enabled the determination of the sizes of optimal constant-composition codes of weight three (and specified distance), leaving only four cases undetermined. Previously, the sizes of constant-composition codes of weight three were known only for those of sufficiently large length.Comment: 13 pages, 1 figure, 4 table

    Spectrum of Sizes for Perfect Deletion-Correcting Codes

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    One peculiarity with deletion-correcting codes is that perfect tt-deletion-correcting codes of the same length over the same alphabet can have different numbers of codewords, because the balls of radius tt with respect to the Levenshte\u{\i}n distance may be of different sizes. There is interest, therefore, in determining all possible sizes of a perfect tt-deletion-correcting code, given the length nn and the alphabet size~qq. In this paper, we determine completely the spectrum of possible sizes for perfect qq-ary 1-deletion-correcting codes of length three for all qq, and perfect qq-ary 2-deletion-correcting codes of length four for almost all qq, leaving only a small finite number of cases in doubt.Comment: 23 page

    Constructions of Optimal and Near-Optimal Multiply Constant-Weight Codes

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    Multiply constant-weight codes (MCWCs) have been recently studied to improve the reliability of certain physically unclonable function response. In this paper, we give combinatorial constructions for MCWCs which yield several new infinite families of optimal MCWCs. Furthermore, we demonstrate that the Johnson type upper bounds of MCWCs are asymptotically tight for fixed weights and distances. Finally, we provide bounds and constructions of two dimensional MCWCs
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