Limitations of error correcting codes over even fields clarified
On the MDS conjecture in even characteristic
Information Theory
Summary
The authors solved a long-standing question about special error-correcting codes called MDS codes when built over fields with an even number of elements. They showed strict limits on how long these codes can be depending on their size, except for two special cases. This result confirms a theory about these codes that had been unproven in this setting. Understanding these limits helps in knowing how efficiently information can be protected against errors.
What this means in practice
- •For communication engineers: Design error-correcting codes over fields of characteristic two with guaranteed maximal length constraints.
- •For data storage architects: Assess limitations on redundancy parameters for storage systems using MDS codes over even characteristic fields.
A theory result. No direct application yet.
Authors
Bocong Chen, Jing Huang
Abstract
We prove that every nontrivial linear $[n,k]$ maximum distance separable (MDS) code over a finite field $\F_q$ of characteristic two satisfies $n\le q+1$, except in dimensions $k=3$ and $k=q-1$, where $n\le q+2$. This resolves the linear MDS conjecture in even characteristic.