关于加法码的极大性
On the Maximality of Additive Codes
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中文总结 AI 辅助
研究加法码的极大性问题，将线性码的ABS模型扩展到加法情形，探讨承认扩展的加法码是否必有加法扩展，通过多种情况分析得出一般答案是否定的，并给出反例，还猜想了素数情形下可扩展加法码的性质。
中文摘要 AI 辅助
一个加法\((n,k,d)_{q^m/q}\)码是\(\mathrm{GF}(q^m)^n\)中\(\mathrm{GF}(q)\)维数为\(km\)且最小汉明距离为\(d\)的\(\mathrm{GF}(q)\)线性子空间。我们首先将线性码的奥尔德森 - 布鲁恩 - 西尔弗曼（ABS）模型扩展到加法情形：长度为\(n\)且在大小为\(q^m\)的字母表上有\(q^{km}\)个码字的码，当且仅当它等价于一个非退化加法码时才承认一个ABS模型。然后我们问一个承认扩展的加法码是否必须承认一个\(\emph{加法}\)扩展。对于线性码（\(m = 1\)）这是奥尔德森和加奇的一个定理。我们将不承认加法扩展的加法码表征为其相关的射影平坦系统是完备的那些码，并且我们证明对于\((n,2,d)_{9/3}\)、\((n,2,d)_{4/2}\)和\((n,3,d)_{4/2}\)码，上述问题的答案再次是肯定的。与线性情形相反，我们表明一般答案是否定的。对于每个平方\(q\)，分散线性集产生可扩展的加法\((n,2,d)_{q^2/q}\)码且不承认加法扩展。此外，一种不同的方法产生一个可扩展的加法\((30,2,24)_{8/2}\)码且不承认加法扩展。因此，对于适当的加法码，相关射影系统的完备性并不意味着码的极大性。我们猜想可扩展的\((n,2,d)_{p^2/p}\)码，\(p\)为素数，总是承认加法扩展。
英文摘要
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 Gács. 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.
