Generalized balanced tournament designs and related codes

Abstract

A generalized balanced tournament design, or a GBTD(k, m) in short, is a (km, k, k − 1)-BIBD defined on a km-set V . Its blocks can be arranged into an m × (km − 1) array in such a way that (1) every element of V is contained in exactly one cell of each column, and (2) every element of V is contained in at most k cells of each row. In this paper, we present a new construction for GBTDs and show that a GBTD(4, m) exists for any integer m ≥ 5 with at most eight possible exceptions. A link between a GBTD(k, m) and a near constant composition code is also mentioned. The derived code is optimal in the sense of its size.

Generalized Balanced Tournament Packings and Optimal Equitable Symbol Weight Codes for Power Line Communications

“…Since there are at most λ blocks containing both x and y, and that no two such blocks can occur in the same column of the GBTP λ (K; v, m × n), the distance between c(x) and c(y) is at least n − λ. We note that the correspondence between GBTPs and ESWCs was observed by Yin et al [33,Theorem 2.2]. However, in the latter paper, the class of codes constructed is called near-constant-composition codes (NCCCs).…”

“…The following summarizes the state-of-the-art results on the existence of GBTD k−1 (k, m). Theorem 3.5 (Lamken [19][20][21][22], Yin et al [33], Chee et al [9], Dai et al [11]). The following holds.…”

“…However, it follows the general strategy of the previous work [11,21,33]. This section outlines the general strategy used, and introduces some required combinatorial designs.…”

“…Our main tools are starters and the method of differences . Starter-adder constructions are ubiquitous in the constructions for GBTDs with index k − 1, associated frames, and other types of similar designs (see, e.g., [9,11,21,22,33]). Unlike previous work and due to the lack of symmetry in our arrays, we fix the positions of the starters in our arrays and "develop" the blocks in a variety of "directions" (see .…”

“…GBTDs have been extensively studied [9,11,[20][21][22]33] and are useful in the constructions of resolvable, near-resolvable, doubly resolvable, and doubly near-resolvable balanced incomplete block designs [20,23,24]. Using the classical correspondence given by Semakov and Zinoviev [29] (see also [4,12,33]), we construct optimal families of ESWCs from certain families of GBTPs. We establish existence results for these families of GBTPs by borrowing standard recursion and direct construction methods from combinatorial design theory.…”