Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Luo, Gaojun, Cao, Xiwang, Li, Shitao, Li, Yang
Institution: Ministry of Industry and Information Technology
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G862364771, G8885165659
OpenAlex ID: W7223206623