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

TL;DR

This paper precisely determines the minimum distance of a new class of constacyclic codes over finite fields, resolving an open problem and analyzing scalar-residue layers of Reed--Muller codes.

Contribution

It proves the exact minimum distance for intermediate constacyclic codes and characterizes the minimum affine support of scalar-residue layers in Reed--Muller codes.

Findings

Exact minimum distance formulas for constacyclic codes in the intermediate range.

Resolution of an open problem posed by Sun, Ding, and Wang.

Characterization of the minimum affine support for scalar-residue 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.