Projective systems and bounds on the length of codes of non-zero defect

We derive bounds on the lengths of linear codes with fixed Singleton defect $s$, working within the framework of projective systems as advocated by Tsfasman and Vlǎduţ.

This geometric perspective allows us to unify and extend a range of existing results.

We introduce the parameter $m^s(k,q)$, denoting the maximum length of a non-degenerate $[n,k,d]_q$ A$^s$MDS code, and more generally $m^s_t(k,q)$, where the dual code is additionally required to be A$^t$MDS.

We also study $\kappa(s,q)$, the maximum dimension $k$ for which a length-maximal A$^s$MDS code exists.

Among our main results, we provide sufficient conditions on $n$ and $k$ under which the dual of an A$^s$MDS code is necessarily A$^s$MDS, addressing a gap in the existing literature.

We show that codes of sufficient length must be projective, meet the Griesmer bound, and be dual to an AMDS code.

Our bounds subsume or improve several results in the literature.

Two conjectures on the non-existence of length-maximal codes of dimension $k\ge 5$ are proposed, supported by computational evidence.