Linearly Embeddable Designs

TL;DR

This paper investigates the conditions under which residual designs are linearly embeddable over finite fields, providing both sufficient and necessary conditions, and applies these results to classical affine designs and Hamada's conjecture.

Contribution

It establishes new criteria for linear embeddability of residual designs and demonstrates their application to classical affine designs and counterexamples to Hamada's conjecture.

Findings

Sufficient condition for linear embeddability based on minimum distance of associated codes.

Residual designs of known classes are linearly embeddable under the new criteria.

Counterexamples to Hamada's conjecture are identified via embeddings of residual designs.

Abstract

A residual design ${\cal{D}}_B$ with respect to a block $B$ of a given design $\cal{D}$ is defined to be linearly embeddable over $GF(p)$ if the $p$-ranks of the incidence matrices of ${\cal{D}}_B$ and $\cal{D}$ differ by one. A sufficient condition for a residual design to be linearly embeddable is proved in terms of the minimum distance of the linear code spanned by the incidence matrix, and this condition is used to show that the residual designs of several known infinite classes of designs are linearly embeddable. A necessary condition for linear embeddability is proved for affine resolvable designs and their residual designs. As an application, it is shown that a residual design of the classical affine design of the planes in $AG(3,2^2)$ admits two nonisomorphic embeddings over $GF(2)$ that give rise to the only known counter-examples of Hamada's conjecture over a field of non-prime order.