由超图产生的加性码

中文摘要 AI 辅助

我们通过与码相关联的整数多拟阵来研究加性码的临界指数。在此背景下，我们给出了Whittle临界定理的一个编码理论证明，给出了临界指数在$h$-射影系统方面的几何描述，并给出了一般界，包括Kung围长界的一个类比。然后我们研究其多拟阵为超图$H$的超图多拟阵的加性码。对于这些码，临界指数被证明由$H$的弱色数决定。如果码是忠实的，则对偶码的最小折叠汉明重量等于$H$的Berge围长。如果$H$是连通的，则最小距离等于$H$的边连通度。作为推论，对于$h\geq2$，我们确定了所有具有连通$H$且达到Singleton界的此类码，即所有忠实的超图加性准MDS码。我们将Griesmer界和线性规划界专门化到超图码，从$H$的加权$2$-截面的拉普拉斯特征值导出了最小距离的下界，并在计算上比较了所有这些界。

英文摘要

We study the critical exponent of additive codes through an integer polymatroid associated with the code. We give a coding-theoretic proof of Whittle's Critical Theorem in this setting, a geometric description of the critical exponent in terms of $h$-projective systems, and general bounds, including an analogue of Kung's girth bound. We then study additive codes whose polymatroid is the hypergraphic polymatroid of a hypergraph $H$. For these codes the critical exponent turns to be determined by the weak chromatic number of $H$. If the code is faithful, then the minimum folded Hamming weight of the dual code is equal to the Berge girth of $H$. If $H$ is connected, the minimum distance is equal to the edge-connectivity of $H$. As a consequence, for $h\geq2$ we determine all such codes with connected $H$ that attain the Singleton bound, that is, all faithful hypergraphic additive quasi-MDS codes. We specialise the Griesmer and linear programming bounds to hypergraphic codes, we derive a lower bound on the minimum distance from the Laplacian eigenvalues of the weighted $2$-section of $H$, and we compare all these bounds computationally.

发表机构

• Université de Rennes（雷恩大学）

机构由 AI 辅助整理，请以论文原文为准。