Search-to-Decision Reductions for the Linear and General Code Equivalence Problems

2026-08-11

AI Summary

This work addresses the challenge of efficiently reducing the search version to the decision version for Linear Code Equivalence (LCE) and Generalized Code Equivalence (GCE). Inspired by recent advances in permutation code equivalence reductions, it proposes the first deterministic polynomial-time search-to-decision reduction framework applicable to both LCE and GCE. The approach leverages a decision oracle to recover the permutation component and employs the Engel–Schneider algorithm to efficiently reconstruct the diagonal scaling and field automorphism components, thereby fully recovering the equivalence transformation. This contribution not only extends the applicability of classical reduction techniques in coding theory but also significantly enhances the computational efficiency of solving LCE and GCE problems.

Abstract

In this paper, we present efficient search-to-decision reductions for the Linear Code Equivalence (LCE) and Generalised Code Equivalence (GCE) problems. Our methodology is inspired by the recent search-to-decision reduction for Permutation Code Equivalence. We demonstrate how to recover the permutation component of the equivalence using a decision oracle, and subsequently show one way of recovering the diagonal and field automorphism components in deterministic polynomial time by leveraging the elegant Engel-Schneider algorithm for diagonal equivalence.

Problem

• Linear Code Equivalence
• Generalised Code Equivalence
• search-to-decision reduction
• code equivalence

Innovation

• search-to-decision reduction
• Linear Code Equivalence
• Generalised Code Equivalence
• Engel-Schneider algorithm
• polynomial-time recovery