Intermediate Constacyclic Codes and Scalar-Residue Reed--Muller Layers

Abstract

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.