# Group divisible designs with block size four and type g^u m^1 - III

## TL;DR

This paper establishes the existence of certain group divisible designs with block size four and specific group types, expanding the known cases where necessary conditions are also sufficient.

## Contribution

It proves the sufficiency of necessary conditions for 4-GDDs of type g^u m^1 for a wide range of g values, including many composite and power-of-two cases.

## Key findings

- Necessary conditions are sufficient for g = 14, 20, 22, 26, 28, ... , 496.
- Sufficiency is shown for g = 2^t q^s with specific primes q.
- Possible exceptions include certain large composite g values.

## Abstract

We deal with group divisible designs that have block size 4 and group type g^u m^1, where g = 2 or 4 (mod 6). We show that the necessary conditions for the existence of a 4-GDD of type g^u m^1 are sufficient when g = 14, 20, 22, 26, 28, 32, 34, 38, 40, 44, 46, 50, 52, 58, 62, 68, 76, 88, 92, 100, 104, 116, 124, 136, 152, 160, 176, 184, 200, 208, 224, 232, 248, 272, 304, 320, 368, 400, 448, 464 and 496. Using these results we go on to show that the necessary conditions are sufficient for g = 2^t q^s, q = 19, 23, 25, 29, 31, s, t = 1, 2, ..., as well as for g = 2^t q, q = 2, 5, 7, 11, 13, 17, t = 1, 2, ..., with possible exceptions 56^9 m^1, 80^9 m^1 and 112^9 m^1 for a few large values of m.

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Source: https://tomesphere.com/paper/1903.07064