Generalized ovals, 2.5-dimensional additive codes, and multispreads

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 $q^h+1$ if $q$ is odd. Those geometric objects are called generalized ovals. We show that cardinality $q^h+2$ 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

💡 Research Summary

The paper investigates additive codes over finite fields through their geometric counterpart, namely projective systems, and uses this correspondence to construct new combinatorial objects and improve code parameters. After recalling the classical result that in PG(2,q) the maximum number of (h‑1)-dimensional subspaces with no three lying in a hyperplane is q^h + 1 for odd q (the so‑called generalized ovals), the authors show that by reducing the ambient dimension one can achieve q^h + 2 subspaces. This extension of generalized ovals is proved in Proposition 1 and Theorem 4, and a more general bound is given in Theorem 2.

The paper then formalises the link between additive codes and projective h‑systems: an additive