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.
Categories: math.CO cs.IT math.IT