Generalized perfect difference families and their application to variable-weight geometric orthogonal codes

TL;DR

This paper investigates the existence of generalized perfect difference families (PDFs) related to geometric orthogonal codes, providing complete solutions for certain parameters and constructing new variable-weight GOCs.

Contribution

It establishes existence conditions for generalized PDFs with specific parameters using auxiliary designs and recursive methods, advancing the theory of geometric orthogonal codes.

Findings

Existence of generalized (n×m, {3,4}, 1)-PDF if and only if nm ≡ 1 mod 6.

Complete solutions for the existence of generalized (n×m, {3,4,5}, 1)-PDF.

Construction of variable-weight perfect (n×m, K, 1)-GOCs.

Abstract

Motivated by the application in geometric orthogonal codes (GOCs), Wang et al. introduced the concept of generalized perfect difference families (PDFs), and established the equivalence between GOCs and a certain type of generalized PDFs recently. Based on the relationship, we discuss the existence problem of generalized $(n\times m,K,1)$-PDFs in this paper. By using some auxiliary designs such as semi-perfect group divisible designs and several recursive constructions, we prove that a generalized $(n\times m, \{3,4\}, 1)$-PDF exists if and only if $nm\equiv1\pmod{6}$. The existence of a generalized $(n\times m, \{3,4,5\}, 1)$-PDF is also completely solved possibly except for a few values. As a consequence, some variable-weight perfect $(n\times m,K,1)$-GOCs are obtained.\vspace{0.2cm} {\bf Keywords}: generalized perfect difference family, generalized perfect difference packing, geometric orthogonal code, semi-perfect group divisible design