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

TL;DR

This work provides an explicit, deterministic construction of rate-$\frac{1}{2}$ Wozencraft ensemble codes over $\mathbb{F}_q$ achieving minimum distance $\Omega(\sqrt{k})$ by using Bose-Chowla Sidon sets and a cyclotomic framework in $\mathbb{F}_{q^{k}}$, under the assumption that $k+1$ is prime and $q$ is a primitive root modulo $k+1$ (supported for infinitely many $k$ by Artin's conjecture). It then shows how to puncture these codes to obtain higher rates $r>\frac{1}{2}$ with distance $\Omega_{r}(\sqrt{k}) = \Omega\big((1-\sqrt{2-1/r})\sqrt{k}\big)$, preserving explicitness and polynomial-time computability. The core technique combines a weight-analysis in a ring extension with Sidon-set combinatorics to ensure many coefficient positions are uniquely determined, yielding strong lower bounds on distance. The results advance explicit constructions that approach GV-type guarantees and point to further opportunities in reducing ensemble size while maintaining substantial distance gains, with links to Weldon-type constructions for low-rate Zyablov-bound regimes.

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 $Ω(\sqrt{k})$ 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 $q$.

Theorems & Definitions (37)

• Theorem 1.1: Informal
• Conjecture 2.1: Artin's Conjecture on Primitive Roots
• Remark 2.2
• Definition 2.3
• Proposition 2.4
• proof
• Definition 2.5
• Remark 2.6
• Theorem 2.7: Bose-Chowla, bose1960theorems
• Theorem 3.1