A construction of group divisible designs with zero block sum

TL;DR

This paper presents a new method for constructing group divisible designs over binary extension fields with various block sizes, inspired by decoding binary quadratic residue codes, and proposes a conjecture for larger block sizes.

Contribution

It introduces a novel construction of group divisible designs on binary extension fields for block sizes 3 to 7, motivated by coding theory, and suggests a conjecture for larger sizes.

Findings

Constructed group divisible designs with block sizes 3 to 7.

Motivated by decoding binary quadratic residue codes.

Proposed a conjecture for larger block sizes.

Abstract

This paper gives a construction of group divisible designs on the binary extension fields with block sizes 3, 4, 5, 6, and 7, respectively, which is motivated from the decoding of binary quadratic residue codes. A conjecture is proposed for this construction of group divisible designs with larger block sizes.