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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.17022 |
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| _version_ | 1866911725267189760 |
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| author | Yang, Yaoran Zhang, Yutong |
| author_facet | Yang, Yaoran Zhang, Yutong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_17022 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Intermediate Constacyclic Codes and Scalar-Residue Reed--Muller Layers Yang, Yaoran Zhang, Yutong Information Theory 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. |
| title | Intermediate Constacyclic Codes and Scalar-Residue Reed--Muller Layers |
| topic | Information Theory |
| url | https://arxiv.org/abs/2605.17022 |