The nonexistence of quaternary linear codes with parameters [243,5,181], [248,5,185]\ and [240,5,179] (Q2761042)

Denote by \(n_4(k,d)\) the least \(n\) for which a quaternary \([n,k,d]\) code exists. It has been known that for each \(d\in \{179,181,182,185\}\) there exists an integer \(g(d)\) with \(n_4(5,d)\in \{g(d),g(d)+1\}\). The author proves \(n_4(5,d)=g(d)+1\) for all four values of \(d\). The integer \(g(d)\) corresponds to the Griesmer bound, and so the non-existence of the quaternary \([g(d),5,d]\)-code can be achieved by (quite lengthy) combinatorial arguments in \(PG(4,4)\) using a correspondence between codes meeting the Griesmer bound and minihypers in \(PG(4,4)\).