On the maximum length of cyclic MDS codes

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 $\mathrm{\ell }\ge 11$ with $(\mathrm{\ell }-1)\mathrm{/}2$ 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.