Classification of $3 \operatorname{mod} 5$ arcs in $\operatorname{PG}(3,5)$

TL;DR

This paper completes the classification of (3 mod 5)-arcs in PG(3,5), uncovers exceptional examples that disprove a conjecture, and extends the proof regarding the non-existence of certain Griesmer codes.

Contribution

It provides a full classification of (3 mod 5)-arcs in PG(3,5), identifies counterexamples to a prior conjecture, and extends the proof of the non-existence of specific Griesmer codes.

Findings

Complete classification of (3 mod 5)-arcs in PG(3,5)

Discovery of two exceptional counterexamples

Extended proof of non-existence of Griesmer [104, 4, 82]_5-codes

Abstract

The proof of the non-existence of Griesmer $[104, 4, 82]_5$-codes is just one of many examples where extendability results are used. In a series of papers Landjev and Rousseva have introduced the concept of $(t\operatorname{mod} q)$-arcs as a general framework for extendability results for codes and arcs. Here we complete the known partial classification of $(3 \operatorname{mod} 5)$-arcs in $\operatorname{PG}(3,5)$ and uncover two missing, rather exceptional, examples disproving a conjecture of Landjev and Rousseva. As also the original non-existence proof of Griesmer $[104, 4, 82]_5$-codes is affected, we present an extended proof to fill this gap.