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

Abstract: In their 2007 book, Tsfasman and Vl\v{a}du\c{t} 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 $\kappa(s,q)$, 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.