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

翻译：我们基于加法码的几何对应物——即射影系统——提出了有限域上加法码的构造与界。已知在PG$(2,q)$中，使得任意超平面不包含其中三个的$(h-1)$-空间的最大数目为$q^h+1$（当$q$为奇数时）。这类几何对象被称为广义椭圆。我们证明，若略微降低维度，则可能达到基数$q^h+2$。我们完全确定了GF$(9)$上维度为$2.5$的加法码的最小可能长度，并对其他小参数给出了改进的构造，包括性能优于最佳线性码的码。作为应用，我们考虑了PG$(4,q)$中的多重散布，特别地，完成了GF$(4)$-线性$64$元单重码参数的表征。关键词：加法码，射影系统，广义椭圆，多重散布，单重码，双重码