A new reducibility results for minihypers in finite projective geometries

TL;DR

The paper develops a new Reducibility Theorem for $(n,w)$-minihypers in finite projective geometries over a prime $p$, showing that under specified congruence conditions a minihyper must decompose as $\mathcal{F}=\mathcal{F}'+\chi_L$ along a unique line $L$. It then uses this tool to obtain a precise classification of $(70,22)$-minihypers in $\operatorname{PG}(4,3)$, showing they are either the sum of a solid and a $(30,9)$-minihyper or the sum of a $(66,21)$-minihyper and a line, with the latter case informed by lower-dimensional classifications. These results connect finite-geometry structures with coding-theory existence problems, providing new insight into the existence of certain ternary Griesmer codes of dimension $6$ and advancing arc/minihyper theory via modular and projection techniques. The methods blend divisibility arguments for hyperplane counts, projections, and the arc–code correspondence to derive structural decompositions and classifications.

Abstract

In this paper we prove a new reducibility result for mini-hypers in projective geometries over finite fields. It is further used to characterize the minihypers with parameters (70, 22) in PG(4, 3). The latter can be used to attack the existence problem for some hypothetical ternary Griesmer codes of dimension 6.

Theorems & Definitions (26)

• Lemma 2.1
• Lemma 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 3.1
• proof
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3