Group divisible designs with block size 4 and type g^u m^1 - II

TL;DR

This paper establishes the existence conditions for 4-group divisible designs with specific parameters, extending known results and completing the spectrum for certain types, thereby advancing combinatorial design theory.

Contribution

It provides new existence results for 4-GDDs of type g^u m^1 under various modular conditions, filling gaps in the existing spectrum.

Findings

Necessary conditions are sufficient for certain g values.

Extended the spectrum for g ≡ 1 or 5 (mod 6).

Completed the spectrum for specific 4-GDD types.

Abstract

We show that the necessary conditions for the existence of 4-GDDs of type g^u m^1 are sufficient for g congruent to 0 (mod h), h = 39, 51, 57, 69, 87, 93, and for g = 13, 17, 19, 23, 25, 29, 31 and 35. More generally, we show that for all g congruent to 3 (mod 6), the possible exceptions occur only when u = 8 and g is not divisible by any of 9, 15, 21, 33, 39, 51, 57, 69, 87 or 93. Consequently we are able to extend the known spectrum for g congruent to 1 and g congruent to 5 (mod 6). Also we complete the spectrum for 4-GDDs of type (3a)^4 (6a)^1 (3b)^1.