An algorithm for list decoding number field codes

TL;DR

This paper introduces a polynomial-time list decoding algorithm for algebraic number field codes, providing the first explicit decoding procedure for these codes based on advanced lattice reduction and module algorithms.

Contribution

It presents the first explicit polynomial-time list decoding algorithm for algebraic number field codes, building on lattice reduction and module computation techniques.

Findings

Decoding algorithm operates in polynomial time.

First explicit decoding procedure for number field codes.

Utilizes advanced lattice and module algorithms.

Abstract

We present an algorithm for list decoding codewords of algebraic number field codes in polynomial time. This is the first explicit procedure for decoding number field codes whose construction were previously described by Lenstra and Guruswami. We rely on an equivalent of the LLL reduction algorithm for $\OK$-modules due to Fieker and Stehl\'e and on algorithms due to Cohen for computing the Hermite normal form of matrices representing modules over Dedekind domains.