2025/11/19 by Krotov, Denis S., Kurz, Sascha
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Finite Group Theory Research #Information Theory (cs.IT) #graph theory and CDMA systems
paper · doi:10.48550/arxiv.2511.15843
openalex publication_date 2025/11/19 · openalex created_date 2025/11/23 · openalex updated_date 2026/07/28
We present constructions and bounds for additive codes over a finite field in terms of their geometric counterpart, i.e., projective systems. It is known that the maximum number of (h-1)-spaces in PG(2,q), such that no hyperplane contains three, is given by qh+1 if q is odd. Those geometric objects are called generalized ovals. We show that cardinality qh+2 is possible if we decrease the dimension a bit. We completely determine the minimum possible lengths of additive codes over GF(9) of dimension 2.5 and give improved constructions for other small parameters, including codes outperforming the best linear codes. As an application, we consider multispreads in PG(4,q), in particular, completing the characterization of parameters of GF(4)-linear 64-ary one-weight codes. Keywords: additive code, projective system, generalized oval, multispread, one-weight code, two-weight code