Maximum length limits of special cyclic error correcting codes
On the maximum length of cyclic MDS codes
Information Theory
Summary
This paper studies the longest possible size of a type of error-correcting code called cyclic MDS codes, which are used to detect and fix errors in data. The authors found new rules that limit how long these codes can be based on mathematical properties of their size and the field they work over. They use advanced algebra and number theory tools to prove that, under certain conditions, these codes cannot exceed a specific length related to the size of the field. This helps clarify when such codes are possible and when they are not.
What this means in practice
- •For error correction engineers: Determine achievable code lengths when designing cyclic MDS codes for communication systems based on field and length constraints.
- •For cryptography developers: Use length bounds on cyclic MDS codes to assess feasibility of constructing certain cryptographic primitives relying on these codes.
A theory result. No direct application yet.
Authors
Gaojun Luo, Xiwang Cao, Shitao Li, Yang Li
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.