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

Source abstract

We derive bounds on the lengths of linear codes with fixed Singleton defect, working within the framework of projective systems. This geometric perspective allows us to unify and extend a range of existing results. We study the maximum length of a non-degenerate linear code when its dimension, field order, and Singleton defect are fixed, and refine this question by also prescribing the defect of the dual code. We further investigate the maximum dimension in which a code can attain the general length bound. Among our main results, we provide sufficient conditions under which a code and its dual have equal Singleton defect, addressing a gap in the existing literature. We show that codes of sufficient length must be projective, meet the Griesmer bound, and have an almost maximum distance separable dual. Our bounds subsume or improve several results in the literature. For field orders greater than three, we propose two conjectures on length-maximal codes of dimension at least five, supported by computational evidence. Received: 05 May 2025 | Revised: 24 June 2026 | Accepted: 26 June 2026