Projective systems and bounds on the length of codes of non-zero defect
Tim L. Alderson, Zhipeng Zhang
TL;DR
By reframing linear codes as projective systems, the paper derives tight bounds on the maximum lengths of non-zero Singleton-defect codes $m^{s}(k,q)$ and their dual-constraints $m^{s}_{t}(k,q)$, introducing the central parameter $\kappa(s,q)$ that captures maximal dimensions for maximal arcs. The main approach leverages projective-geometry techniques, quotient constructions, and shortening to connect $A^s$MDS codes with arcs and caps in PG-spaces, yielding unified bounds that subsume many prior results. It also provides sufficient conditions under which the dual of an $A^s$MDS code remains in the same defect class and investigates when long codes must be NMDS, accompanied by conjectures on $\kappa(s,q)$. Overall, the work advances understanding of code lengths under Singleton defect constraints and duality, with potential impacts on the design of long error-correcting codes over finite fields.
Abstract
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.