Constructions and Applications of Perfect Difference Matrices and Perfect Difference Families

TL;DR

This paper develops new recursive methods to construct perfect difference matrices and families, significantly expanding their existence range and enabling applications in optical orthogonal codes and radar arrays.

Contribution

It introduces novel recursive constructions for PDM(3,m)s, proving their existence for all odd m<1000 except for known and potential exceptions, and fully characterizes certain perfect difference families.

Findings

PDM(3,m)s exist for all odd m<1000 except 9, 11, and possibly 33.

Complete classification of (g, {3,4}, 1)-PDFs with block size ratio ≥ 1/14.

Construction of perfect strict optical orthogonal codes with weights 3 and 4.

Abstract

Perfect difference families (PDFs for short) are important both in theoretical and in applications. Perfect difference matrices (PDMs for short) and the equivalent structure had been extensively studied and used to construct perfect difference families, radar array and related codes. The necessary condition for the existence of a PDM$(n,m)$ is $m\equiv 1\pmod2$ and $m\geq n+1$. So far, PDM$(3,m)$s exist for odd $5\leq m\leq 201$ with two definite exceptions of $m=9,11$. In this paper, new recursive constructions on PDM$(3,m)$s are investigated, and it is proved that there exist PDM$(3,m)$s for any odd $5\leq m<1000$ with two definite exceptions of $m=9,11$ and $33$ possible exceptions. A complete result of $(g,\{3,4\},1)$-PDFs with the ratio of block size $4$ no less than $\frac{1}{14}$ is obtained. As an application, a complete class of perfect strict optical orthogonal codes with weights $3$ and $4$ is obtained.