Summary
Some special codes help computers detect and fix errors in data. There was a prediction by Ward about certain patterns in these codes, called Griesmer codes. The authors have now proven this prediction correct, meaning we understand these codes better. This helps show exactly how these codes behave with certain powers of a prime number.
What this means in practice
- •For error correction engineers: Confirm exact divisibility properties of Griesmer codes when designing reliable communication systems.
- •For cryptographers: Use the proven divisibility results to analyze code-based cryptographic schemes relying on Griesmer codes.
A theory result. No direct application yet.
Abstract
Let $C$ be an $[n,k,d]_q$ Griesmer code, where $q=p^f$ with $p$ prime, and write $e=ν_p(d)$. Ward conjectured that $C$ is $p^{\max\{0,e-f+1\}}$-divisible. In this paper, we prove this conjecture.