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

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

The 3-GDDs of type g3u2g^3u^2

A 3-GDD of type ${g^3u^2}$ exists if and only if $g$ and $u$ have the same parity, $3$ divides $u$ and $u\leq 3g$.Such a 3-GDD of type ${g^3u^2}$ is equivalent to an edge decomposition of $K_{g,g,g,u,u}$ into triangles

Further results on nearly Kirkman triple systems with subsystems

AbstractIn this paper we further discuss the embedding problem for nearly Kirkman triple systems and get the result that: (1) For u≡v≡0(mod6), v⩾78, and u⩾3.5v, there exists an NKTS(u) containing a sub-NKTS(v). (2) For v=18,24,30,36,42,48,54,60,66 or 72, there exists an NKTS(u) containing a sub-NKTS(v) if and only if u≡0(mod6) and u⩾3v

Further combinatorial constructions for optimal frequency-hopping sequences

AbstractFrequency-hopping multiple-access (FHMA) spread-spectrum communication systems employing multiple frequency shift keying as data modulation technique were investigated by Fuji-Hara, Miao and Mishima [R. Fuji-Hara, Y. Miao, M. Mishima, Optimal frequency hopping sequences: A combinatorial approach, IEEE Trans. Inform. Theory 50 (2004) 2408–2420] from a combinatorial approach, where a correspondence between frequency-hopping (FH) sequences and partition-type cyclic difference packings was established, and several combinatorial constructions were provided for FHMA systems with a single optimal FH sequence. In this paper, by means of this correspondence, we describe more combinatorial constructions for such optimal FH sequences. As a consequence, more new infinite series of optimal FH sequences are obtained

Existence of frame SOLS of type anb1

AbstractAn SOLS (self-orthogonal latin square) of order v with ni missing sub-SOLS (holes) of order hi (1⩽i⩽k), which are disjoint and spanning (i.e. ∑i=1knihi=v), is called a frame SOLS and denoted by FSOLS(h1n1h2n2 ⋯hknk). It has been proved that for b⩾2 and n odd, an FSOLS(anb1) exists if and only if n⩾4 and n⩾1+2b/a. In this paper, we show the existence of FSOLS(anb1) for n even and FSOLS(an11) for n odd