Cyclic (v;r,s;lambda) difference families with two base blocks and v <= 50

TL;DR

This paper constructs numerous new cyclic difference families with parameters up to v=50, solving open problems and deriving combinatorial designs like BIBDs and Golay pairs, along with new D-optimal designs.

Contribution

It introduces new cyclic difference families with specific parameters, including solutions to previously open existence questions, and proposes a normal form for classification.

Findings

Constructed difference families with parameters (45;22,22;21) and (50;25,20;20).

Derived a BIBD with v=45, b=90, r=44, k=22, lambda=21.

Produced nine new D-optimal designs.

Abstract

We construct many new cyclic (v;r,s;lambda) difference families with v less than or equal 50. In particular we construct the difference families with parameters (45;18,10;9), (45;22,22;21), (47;21,12;12), (47;19,15;12), (47;22,14;14), (48;20,10;10), (48;24,4;12), (50;25,20;20) for which the existence question was an open problem. The (45;22,22;21) difference family gives a BIBD with parameters v=45, b=90, r=44, k=22 and lambda=21, and the one with parameters (50;25,20;20) gives a pair of binary sequences of length 50 with zero periodic autocorrelation function (the periodic analog of a Golay pair). We also construct nine new D-optimal designs. A normal form for cyclic difference families is proposed and used effectively in compiling the list of known and new difference families.