235 research outputs found

    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

    A Pair of Disjoint 3-GDDs of type g^t u^1

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    Pairwise disjoint 3-GDDs can be used to construct some optimal constant-weight codes. We study the existence of a pair of disjoint 3-GDDs of type gtu1g^t u^1 and establish that its necessary conditions are also sufficient.Comment: Designs, Codes and Cryptography (to appear

    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

    Fixed block configuration group divisible designs with block size six

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    AbstractWe present constructions and results about GDDs with two groups and block size six. We study those GDDs in which each block has configuration (s,t), that is in which each block has exactly s points from one of the two groups and t points from the other. We show the necessary conditions are sufficient for the existence of GDD(n,2,6;λ1,λ2)s with fixed block configuration (3,3). For configuration (1,5), we give minimal or near-minimal index examples for all group sizes n≥5 except n=10,15,160, or 190. For configuration (2,4), we provide constructions for several families of GDD(n,2,6;λ1,λ2)s

    Group divisible designs of four groups and block size five with configuration (1; 1; 1; 2)

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    We present constructions and results about GDDs with four groups and block size five in which each block has Configuration (1,1,1,2)(1, 1, 1, 2), that is, each block has exactly one point from three of the four groups and two points from the fourth group. We provide the necessary conditions of the existence of a GDD(n,4,5;λ1,λ2)(n, 4, 5; \lambda_1, \lambda_2) with Configuration (1,1,1,2)(1, 1, 1, 2), and show that the necessary conditions are sufficient for a GDD(n,4,5;λ1,(n, 4, 5; \lambda_1, λ2)\lambda_2) with Configuration (1,1,1,2)(1, 1, 1, 2) if n≢0(n \not \equiv 0 (mod 6)6), respectively. We also show that a GDD(n,4,5;2n,6(n−1))(n, 4, 5; 2n, 6(n - 1)) with Configuration (1,1,1,2)(1, 1, 1, 2) exists, and provide constructions for a GDD(n=2t,4,5;n,3(n−1))(n = 2t, 4, 5; n, 3(n - 1)) with Configuration (1,1,1,2)(1, 1, 1, 2) where n≠12n \not= 12, and a GDD(n=6t,4,5;4t,2(6t−1))(n = 6t, 4, 5; 4t, 2(6t - 1)) with Configuration (1,1,1,2)(1, 1, 1, 2) where n≠6n \not= 6 and 1818, respectively

    Fixed block configuration GDDs with block size 6 and (3, r)-regular graphs

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    Chapter 1 is used to introduce the basic tools and mechanics used within this thesis. Most of the definitions used in the thesis will be defined, and we provide a basic survey of topics in graph theory and design theory pertinent to the topics studied in this thesis. In Chapter 2, we are concerned with the study of fixed block configuration group divisible designs, GDD(n; m; k; λ1; λ2). We study those GDDs in which each block has configuration (s; t), that is, GDDs in which each block has exactly s points from one of the two groups and t points from the other. Chapter 2 begins with an overview of previous results and constructions for small group size and block sizes 3, 4 and 5. Chapter 2 is largely devoted to presenting constructions and results about GDDs with two groups and block size 6. We show the necessary conditions are sufficient for the existence of GDD(n, 2, 6; λ1, λ2) with fixed block configuration (3; 3). For configuration (1; 5), we give minimal or nearminimal index constructions for all group sizes n ≥ 5 except n = 10, 15, 160, or 190. For configuration (2, 4), we provide constructions for several families ofGDD(n, 2, 6; λ1, λ2)s. Chapter 3 addresses characterizing (3, r)-regular graphs. We begin with providing previous results on the well studied class of (2, r)-regular graphs and some results on the structure of large (t; r)-regular graphs. In Chapter 3, we completely characterize all (3, 1)-regular and (3, 2)-regular graphs, as well has sharpen existing bounds on the order of large (3, r)- regular graphs of a certain form for r ≥ 3. Finally, the appendix gives computational data resulting from Sage and C programs used to generate (3, 3)-regular graphs on less than 10 vertices

    α-Resolvable λ-fold G-designs

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    A λ-fold G-design is said to be α-resolvable if its blocks can be partitioned into classes such that every class contains each vertex exactly α times. In this paper we study the existence problem of an α-resolvable λ-fold G-design oforder v in the case when G is any connected subgraph of K_4 and prove that the necessary conditions for its existence are also sufficient
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