Two series of equitable symbol weight codes meeting the Plotkin bound

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

“…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 .…”

“…GBTPs extend the concept of generalized balanced tournament designs (GBTDs) introduced by Lamken and Vanstone . GBTDs have been extensively studied and are useful in the constructions of resolvable, near‐resolvable, doubly resolvable, and doubly near‐resolvable balanced incomplete block designs . Using the classical correspondence given by Semakov and Zinoviev (see also ), we construct optimal families of ESWCs from certain families of GBTPs.…”

“…Authority m Corollary 6.10 {s : 10 ≤ s ≤ 57} \ {16, 20,24,28,32,36,40,44,48, 50, 52, 54, 55, 56} Lemma 7.4 with (n, g) ∈ {(10, 0), (11,0), (12,0), (13,0), (11,10), (11,11), (14, implies that n 2 − n 1 > 3 for all n 2 ∈ E and n 2 > n 1 . However, a quick check on E gives a contradiction.…”