Exact Minimum Distance of the Ding--Li--Xia Cyclic Codes

Yutong Zhang; Yaoran Yang

Exact Minimum Distance of the Ding--Li--Xia Cyclic Codes

Information Theory 2026-07-27 v1

Authors: Yutong Zhang , Yaoran Yang

Abstract

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.

Keywords

cyclic codes minimum distance ding-li-xia codes bch bound projective subspace subspace power sum

Cite

@article{arxiv.2607.24646,
  title  = {Exact Minimum Distance of the Ding--Li--Xia Cyclic Codes},
  author = {Yutong Zhang and Yaoran Yang},
  journal= {arXiv preprint arXiv:2607.24646},
  year   = {2026}
}