AI 中文总结
研究丁 - 李 - 夏循环码$\mho(q,m,h)$的精确最小距离，通过显式射影子空间构造及相关引理和估计，证明对于特定的$q$、$m$、$h$，其最小距离为$(q^{h + 1}-1)/(q - 1)$，构造的码字达BCH下界确定精确距离。
AI 中文摘要
丁、李和夏引入的循环码$\mho(q,m,h)$构成了删余二元里德 - 穆勒码的非二元推广。他们建立了界$(q^{h + 1}-1)/(q - 1)\leq d(\mho(q,m,h))\leq 2q^h - 1$并询问BCH下界是否总是精确的。本文证明，对于每个素数幂$q$、每个$m\geq 2$以及每个$1\leq h\leq m - 1$，最小距离为$d(\mho(q,m,h))=(q^{h + 1}-1)/(q - 1)$。通过显式射影子空间构造得到上界。对于$\F_{q^m}$的任何$(h + 1)$维$\F_q$子空间$V$，集合$V^{[q - 1]}=\{x^{q - 1}:x\in V\setminus\{0\}\}$支撑一个重量为$(q^{h + 1}-1)/(q - 1)$的码字。其属于$\mho(q,m,h)$可由子空间幂和的消失引理及数字和估计$s_q((q - 1)a)\leq(q - 1)\wtq(a)$得出。构造的码字达到BCH下界，从而确定了精确最小距离。
英文摘要
The cyclic codes $\mho(q,m,h)$ introduced by Ding, Li, and Xia form a nonbinary generalization of punctured binary Reed--Muller codes. Ding, Li, and Xia established the bounds $(q^{h+1}-1)/(q-1)\leq d(\mho(q,m,h))\leq 2q^h-1$ and asked whether the BCH lower bound is always exact. This paper proves that, for every prime power $q$, every $m\geq 2$, and every $1\leq h\leq m-1$, the minimum distance is $d(\mho(q,m,h))=(q^{h+1}-1)/(q-1)$. The upper bound is obtained by an explicit projective-subspace construction. For any $(h+1)$-dimensional $\F_q$-subspace $V$ of $\F_{q^m}$, the set $V^{[q-1]}=\{x^{q-1}:x\in V\setminus\{0\}\}$ supports a codeword of weight $(q^{h+1}-1)/(q-1)$. Its membership in $\mho(q,m,h)$ follows from a vanishing lemma for subspace power sums and the digit-sum estimate $s_q((q-1)a)\leq(q-1)\wtq(a)$. The constructed codeword meets the BCH lower bound and therefore determines the exact minimum distance.
