A 2024 paper of Sun, Ding and Wang introduced a second class of constacyclic codes over finite fields, denoted $C(q,m,r,\ell)$, with length $(q^m-1)/r$, where $r\mid(q-1)$ and the defining monomials have total $q$-ary degree congruent to $r-1$ modulo $r$. In the non-projective intermediate range $2<r<q-1$ the paper gave a sharp-looking upper bound and a BCH-type lower bound, and left the minimum distance open. We prove that the upper bound is the exact minimum distance for every admissible intermediate parameter. More precisely, if $\ell=(q-1)a+b<(q-1)m-1$, $0\le b\le q-2$, and $b\equiv r-1\pmod r$, then, for every prime power $q$, every divisor $r$ of $q-1$ with $2<r<q-1$, and every $m\ge2$, \[ d(C(q,m,r,\ell))= \begin{cases} \displaystyle \frac{q-1}{r}(q-b+1)q^{m-a-2},&0\le a\le m-2,\\[1mm] \displaystyle \frac{q-b+r-2}{r},&a=m-1. \end{cases} \] The first line settles the open problem of Sun, Ding and Wang; the second line is the terminal case already forced by their BCH bound. We also determine the minimum affine support of every non-terminal scalar-residue layer of a generalized Reed--Muller code. The resulting dichotomy says that the first Reed--Muller weight survives exactly for residue classes $0$ and $1$, while every other residue-matched layer starts at the second Reed--Muller weight. The proof uses the hidden scalar homogeneity of the evaluation model, an orbit-counting obstruction for minimum Reed--Muller supports, and a homogeneous pencil construction that attains the second weight.

翻译：Sun、Ding和Wang在2024年的一篇论文中引入了有限域上第二类常循环码，记为$C(q,m,r,\ell)$，其长度为$(q^m-1)/r$，其中$r\mid(q-1)$且定义单项式的总$q$进制次数模$r$同余于$r-1$。在非射影中间范围$2<r<q-1$内，该文给出了一个看似紧的上界和一个BCH型下界，并将最小距离问题留作开放。我们证明，对于每个容许的中间参数，该上界即为精确的最小距离。更精确地说，若$\ell=(q-1)a+b<(q-1)m-1$，$0\le b\le q-2$，且$b\equiv r-1\pmod r$，则对任意素数幂$q$、$q-1$的任意因子$r$（满足$2<r<q-1$）以及任意$m\ge2$，有\[ d(C(q,m,r,\ell))= \begin{cases} \displaystyle \frac{q-1}{r}(q-b+1)q^{m-a-2},&0\le a\le m-2,\\[1mm] \displaystyle \frac{q-b+r-2}{r},&a=m-1. \end{cases} \] 第一行解决了Sun、Ding和Wang的开放问题；第二行是终端情形，已由其BCH界所确定。我们还确定了广义Reed-Muller码每个非终端标量-余子式层的最小仿射支撑集。由此产生的二分性表明：第一个Reed-Muller权值恰好存活于余数类$0$和$1$，而每个其他余数匹配层的起始权值为第二个Reed-Muller权值。证明使用了求值模型中隐藏的标量齐次性、最小Reed-Muller支撑集的轨道计数障碍，以及一个能达到第二个权值的齐次线束构造。