Generalized ovals, 2.5-dimensional additive codes, and multispreads

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Krotov, Denis ; Kurz, Sascha:
Generalized ovals, 2.5-dimensional additive codes, and multispreads.
Bayreuth , 2025 . - 68 S.

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Abstract

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 (l-1)-spaces in PG(2,q), such that no hyperplane contains three, is given by q^l+1 if q is odd. Those geometric objects are called generalized ovals. We show that cardinality q^l+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. 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.