所有高阶二元本原BCH码广义覆盖半径的渐近紧界
Asymptotically Tight Bounds for Generalized Covering Radii of Binary Primitive BCH Codes at All Higher Orders
- Technion – Israel Institute of Technology（以色列理工学院）
- Tel-Hai University of Kiryat Shmona in the Galilee（加利利基里亚特什莫纳泰尔海大学）
- MIGAL – Galilee Research Institute（米加尔-加利利研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对二元本原BCH码的广义覆盖半径问题，提出完备-覆盖框架，给出二阶半径的显式稳定双值界与精确性判据，以及所有高阶的渐近紧上下界，并验证了特定错误数与阶数下的精确结果。
AI中文摘要：
我们研究了长度为$2^m-1$（其中$m$为扩张次数）的二元本原BCH码中，张成若干指定伴随式所需的最少奇偶校验列数。对于四纠错码族，当$m\geq55$时，第二广义覆盖半径恰好为11；当$m\geq16$时，其值为11或12。对于每个固定的错误参数，我们给出了第二半径的显式稳定双值界，以及判定其精确值的算术准则。对于每个$t\geq2$的广义覆盖阶数，我们还得到了显式稳定的上下界。在附加的显式域大小条件下，对于每个固定的$e$，上下界的加性间隙与$t$无关，因此当$t$增大时这些界是渐近紧的。当$2\leq e\leq6$时，上界为自然的公共核计数，且通常与该值的差值为一个与$t$无关的有界校正项。在三阶情况下，这给出了三到六个错误的稳定区间。我们进一步证明，对于三个错误的情况，当$m\geq18$时，每个局限于最高坐标的三维伴随式空间的精确支撑大小为10。一个独立的完备-覆盖框架既支撑了二阶的精确结果，也支撑了所有高阶的渐近紧界。
英文摘要：
We study the generalized covering radii of binary primitive BCH codes, which measure how many parity-check columns suffice to span several prescribed syndromes. For the four-error-correcting family of length $2^m-1$, the second radius is exactly $11$ for $m\geq55$, and it is either $11$ or $12$ for $m\geq16$. For every fixed error parameter at least two, we give an explicit stable bound with two adjacent possible values for the second radius and an arithmetic criterion for exactness. At every higher order, we obtain explicit lower and upper bounds whose additive gap is bounded independently of the order for each fixed error parameter, once explicit field-size conditions hold. The bounds are therefore asymptotically tight as the order grows. The higher-order upper bound retains the common-core support count established by Xiong, Yip, and Zullo; our refinement reduces its sufficient field-size threshold at large order. The refinement combines componentwise mixed degrees with the ordinary degree of a multicone. The upper bounds come from constructing syndrome representations with a common set of parity-check columns.
