Probabilistic verification algorithm for linear codes

TL;DR

This paper introduces a probabilistic algorithm for linear code membership testing that is efficient in time and space, with very low error probability, applicable to any linear code.

Contribution

The paper presents a novel probabilistic algorithm for linear code verification with optimal asymptotic complexity and broad applicability.

Findings

Achieves $O(n ext{log} n)$ time complexity

Uses $O(n^2)$ space complexity

Error probability less than $1/ ext{poly}(n)$

Abstract

In this paper, we propose a probabilistic algorithm suitable for any linear code $C$ to determine whether a given vector $\mathbf{x}$ belongs to $ C$. The algorithm achieves $O(n\log n)$ time complexity, $ O(n^2)$ space complexity and with an error probability less than $1/\mathrm{poly}(n)$ in the asymptotic sense.