A Deterministic Construction of a Large Distance Code from the Wozencraft Ensemble

Abstract

We present an explicit construction of a sequence of rate $1/2$ Wozencraft ensemble codes (over any fixed finite field ${\mathbb{F}}_q$) that achieve minimum distance $\hat{I}\text{c◯}(\sqrt{k})$ where $k$ is the message length. The coefficients of the Wozencraft ensemble codes are constructed using Sidon Sets and the cyclic structure of ${\mathbb{F}}_{q^k}$ where $k+1$ is prime with $q$ a primitive root modulo $k+1$. Assuming Artin's conjecture, there are infinitely many such $k$ for any prime power $q$.