Frame difference families and resolvable balanced incomplete block designs

TL;DR

This paper develops new theoretical results on frame difference families, leading to the construction of resolvable balanced incomplete block designs and applications in coding theory and frequency hopping sequences.

Contribution

It establishes asymptotic existence results for classes of frame difference families and derives new infinite families of resolvable BIBDs with specific parameters.

Findings

New infinite families of 1-rotational RBIBDs over finite fields.

Existence of specific (v,8,1)-RBIBDs for certain v values.

Asymptotic existence of optimal constant composition codes and frequency hopping sequences.

Abstract

Frame difference families, which can be obtained via a careful use of cyclotomic conditions attached to strong difference families, play an important role in direct constructions for resolvable balanced incomplete block designs. We establish asymptotic existences for several classes of frame difference families. As corollaries new infinite families of 1-rotational $(pq+1,p+1,1)$-RBIBDs over $\mathbb{F}_{p}^+ \times \mathbb{F}_{q}^+$ are derived, and the existence of $(125q+1,6,1)$-RBIBDs is discussed. We construct $(v,8,1)$-RBIBDs for $v\in\{624,1576,2976,5720,5776,10200,14176,24480\}$, whose existence were previously in doubt. As applications, we establish asymptotic existences for an infinite family of optimal constant composition codes and an infinite family of strictly optimal frequency hopping sequences.