On existence of Two Classes of Generalized Howell Designs with Block Size Three and Index Two

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ORIGINAL PAPER

On existence of Two Classes of Generalized Howell Designs with Block Size Three and Index Two Jing Shi1 • Jinhua Wang1 Received: 13 July 2019 / Revised: 22 May 2020 Ó Springer Japan KK, part of Springer Nature 2020

Abstract Let t; k; k; s and v be nonnegative integers, and let X be a set of v symbols. A generalized Howell design, denoted t-GHDk ðs; v; kÞ, is an s  s array, each cell of which is either empty or contains a k-set of symbols from X, called a block, such that: (i) each symbol appears exactly once in each row and in each column (i.e. each row and column is a resolution of X); (ii) no t-subset of elements from X appears in more than k cells. A generalized Howell design is a class of doubly resolvable designs , which generalize a number of well-known objects. Particular instances of the parameters correspond to generalized Howell designs are doubly resolvable group divisible designs (DRGDDs). In this paper, we concentrate on the case that t ¼ 2; k ¼ 3 and k ¼ 2, and simply write GHD(s, v; 2). The spectrum of GHDð3n  3; 3n; 2Þ’s and GHDð6n  6; 6n; 2Þ’s is completely established by solving the existence of (3, 2)-DRGDDs of types 3n and 6n . At the same time, we also survey rummage the existence of GHD4 ðn; 4n; 1Þ’s. As their applications, several new classes of multiply constant-weight codes are obtained. Keywords Generalized Howell design  Group divisible design  Doubly resolvable  Frame  Multiply constant-weight code

Mathematics Subject Classification 05B05  05B15  05B30  94B25

Research supported by the National Natural Science Foundation of China under Grant No. 11371207. Electronic supplementary material The online version of this article (https://doi.org/10.1007/s00373020-02198-1) contains supplementary material, which is available to authorized users. & Jinhua Wang [email protected]; [email protected] Jing Shi [email protected] 1

School of Sciences, Nantong University, Nantong 226007, People’s Republic of China

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Graphs and Combinatorics

1 Introduction Let t; k; k; s and v be nonnegative integers, and let X be a set of v symbols. A generalized Howell design, denoted t-GHDk ðs; v; kÞ, is an s  s array, each cell of which is either empty or contains a k-set of symbols from X, called a block, such that: (i) each symbol appears exactly once in each row and in each column (i.e. each row and column is a resolution of X); (ii) no t-subset of elements from X appears in more than k cells. A generalized Howell design is a class of doubly resolvable designs, which generalize a number of well-known objects. It is not only an interesting combination object, but also attracts more and more attention because of its close relationship with permutation arrays [20] and multiply constant-weight codes (see, [12, 23, 38]. We refer the reader to [2, 16] for background on these objects and design theory in general. In this paper we will concentrate on the case when t ¼ 2, in which case we omit these parameters in the notation and simply write GHDðs;

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