On the Maximality of Additive Codes

An additive $(n,k,d)_{q^m/q}$-code is a $\mathrm{GF}(q)$-linear subspace of $\mathrm{GF}(q^m)^n$ of $\mathrm{GF}(q)$-dimension $km$ with minimum Hamming distance $d$. We first extend the Alderson--Bruen--Silverman (ABS) model of linear codes to the additive setting: a code of length $n$ with $q^{km}$ words over an alphabet of size $q^m$ admits an ABS model if and only if it is equivalent to a nondegenerate additive code. We then ask whether an additive code that admits an extension must admit an \emph{additive} extension. For linear codes ($m=1$) this is a theorem of Alderson and G\'acs. We characterize the additive codes admitting no additive extension as those whose associated projective system of flats is complete, and we prove that the answer to the question above is again affirmative for $(n,2,d)_{9/3}$-, $(n,2,d)_{4/2}$-, and $(n,3,d)_{4/2}$-codes. In contrast with the linear case, we show that the answer is negative in general. Scattered linear sets yield, for each square $q$, extendable additive $(n,2,d)_{q^2/q}$-codes admitting no additive extension. Further, a different method yields an extendable additive $(30,2,24)_{8/2}$-code with no additive extension. Consequently, for properly additive codes, completeness of the associated projective system does not imply maximality of the code. We conjecture that extendable $(n,2,d)_{p^2/p}$-codes, $p$ prime, always admit additive extensions.