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

2026/07/27 by Yutong Zhang, Yaoran Yang
#cs.IT #math.IT

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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)≤ d(\mho(q,m,h))≤ 2q^h-1 and asked whether the BCH lower bound is always exact. This paper proves that, for every prime power q, every m≥ 2, and every 1≤ h≤ 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∈ V∖\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)≤(q-1)\wtq(a). The constructed codeword meets the BCH lower bound and therefore determines the exact minimum distance.

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