New upper bound found for lengths of certain error-correcting codes
A Uniform Upper Bound on the Length of MDS Codes from Genus-Two Curves
Information Theory
Summary
This paper finds a new limit on how long special error-correcting codes, called MDS codes from genus-two curves, can be. These codes help fix errors when sending data over noisy channels. The authors show that the length of these codes must be shorter than a specific formula depending on the size of the underlying field. This helps better understand the capabilities and limits of these particular codes in coding theory.
What this means in practice
- •For communication system engineers: Design error-correcting schemes knowing the maximal achievable length constraints of MDS codes from genus-two curves.
- •For cryptography developers: Assess security parameters for protocols using algebraic geometry codes by incorporating tighter length bounds.
A theory result. No direct application yet.
Authors
Haojie Chen
Abstract
We study the maximal length of MDS algebraic geometry codes from genus-two curves over finite fields. Let $C_\mathcal{ L}(D,G)$ be such a code over $\mathbb F_q$, with length $n$ and dimension $k$. Assume that $3\leq k\leq n-3$. We prove that $n<(2q+10\sqrt q+16)/3$. This bound holds for every prime power $q$ and for both parities of $k$. It gives a uniform upper bound of $2q/3+O(\sqrt q)$ for the maximal length. The explicit bound is strictly smaller than $q+1$ when $q\geq125$.