发表机构
Tel-Hai University of Kiryat Shmona in the Galilee; MIGAL—Galilee Research Institute（加利利基里亚特什莫纳泰尔海大学; MIGAL-加利利研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该文证明基于原始BCH码的三角形自由陪集图上的二元全奇偶校验存储码，在固定纠错能力下速率趋于1，并给出双纠错族存储亏缺的精确指数界。
AI 中文摘要
图上的存储码为每个顶点分配一个符号，使得该符号可以从其邻居恢复。我们证明了，对于纠错能力至少为2的每个固定值，基于原始BCH码的三角形自由陪集图上的二元全奇偶校验存储码，其速率趋于1。该结论在显式增长条件下也适用于增加的纠错预算。证明限定了与多项式映射相关的卷积矩阵的二元秩。低坐标阶迫使表示图像的多项式中出现抵消，而多项式秩界将该抵消转化为存储速率的定量估计。对于定义在$\mathbb{F}_{2^m}$上的双纠错族，我们证明了存储亏缺为$\Theta(((1+\sqrt{5})/4)^m)$。这一改进源于多项式系数秩的精确斐波那契公式，以及在有限域求值后匹配的指数下界。
英文摘要
A storage code on a graph assigns a symbol to each vertex so that the symbol can be recovered from its neighbors. We prove that the binary full-parity storage codes on triangle-free coset graphs of primitive BCH codes have rate tending to one for every fixed error-correction capability of at least two. The conclusion also holds for an increasing error budget under an explicit growth condition. The proof bounds the binary rank of convolution matrices associated with polynomial maps. Low coordinate degree forces cancellation in the polynomial representing the image, and a polynomial rank bound converts this cancellation into a quantitative estimate for the storage rate. For the double-error-correcting family over $\mathbb{F}_{2^m}$, we prove that the storage deficiency is $Θ(((1+\sqrt{5})/4)^m)$. This refinement follows from an exact Fibonacci formula for a polynomial coefficient rank and a matching exponential lower bound after finite-field evaluation.
Comments11 pages, no figures
