# Permutation Code Equivalence is not Harder than Graph Isomorphism when Hulls are Trivial

## TL;DR

This paper demonstrates that deciding permutation code equivalence for certain linear codes can be efficiently reduced to graph isomorphism, enabling practical solutions for large codes with trivial hulls.

## Contribution

The paper establishes a deterministic reduction from permutation code equivalence to graph isomorphism for codes with trivial hulls, with an efficient algorithm and experimental validation.

## Key findings

- Permutation code equivalence can be decided in minutes for codes up to length 50,000.
- The reduction relies on computing the inverse of a square matrix related to the codes.
- Experimental results confirm the practical efficiency of the proposed approach.

## Abstract

The paper deals with the problem of deciding if two finite-dimensional linear subspaces over an arbitrary field are identical up to a permutation of the coordinates. This problem is referred to as the permutation code equivalence. We show that given access to a subroutine that decides if two weighted undirected graphs are isomorphic, one may deterministically decide the permutation code equivalence provided that the underlying vector spaces intersect trivially with their orthogonal complement with respect to an arbitrary inner product. Such a class of vector spaces is usually called linear codes with trivial hulls. The reduction is efficient because it essentially boils down to computing the inverse of a square matrix of order the length of the involved codes. Experimental results obtained with randomly drawn binary codes having trivial hulls show that permutation code equivalence can be decided in a few minutes for lengths up to 50,000.