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arXiv · 2610.29810

On the maximum length of cyclic MDS codes

Gaojun Luo·Xiwang Cao·Shitao Li·Yang Li
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Abstract

We study the maximum length of nontrivial simple-root cyclic maximum distance separable (MDS) codes over ${\mathbb F}_q$. We prove $n\le q+1$ if $\gcd(n,q^2-1)>2$, or if the dimension is even and $\gcd(n,q^2-1)>1$. The bound also holds for all lengths whose prime factors lie in $\{2,3,5,7\}$. We obtain further cases involving primes $\ell\ge11$ with $(\ell-1)/2$ prime, under conditions on the exponents and the characteristic. The proofs combine Fourier supports, locality bounds and field descent with arithmetic restrictions on multiplicative orders and prime-adic valuations.

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Gaojun Luo, Xiwang Cao, Shitao Li, Yang Li. 2026-10-21. On the maximum length of cyclic MDS codes. https://arxiv.org/abs/2610.29810

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