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

In their 2007 book, Tsfasman and Vlǎduţ invite the reader to reinterpret existing coding theory results through the lens of projective systems. Redefining linear codes as projective systems provides a geometric vantage point. In this paper, we embrace this perspective, deriving bounds on the lengths of A$^s$MDS codes (codes with Singleton defect $s$). To help frame our discussions, we introduce the parameters $m^{s}(k,q)$, denoting the maximum length of an (non-degenerate) $[n,k,d]_q$ A$^s$MDS code, $m^{s}_t(k,q)$ denoting the maximum length of an (non-degenerate) $[n,k,d]_q$ A$^s$MDS code such that the dual code is an A$^t$MDS code, and $κ(s,q)$, representing the maximum dimension $k$ for which there exists a linear code of (maximal) length $n=(s+1)(q+1)+k-2$. In particular, we address a gap in the literature by providing sufficient conditions on $n$ and $k$ under which the dual of an $[n,k,d]_q$ A$^s$MDS code is also an A$^s$MDS code. Our results subsume or improve several results in the literature. Some conjectures arise from our findings.