Abstract

We present a polynomial time reduction from the multi-graph isomorphism problem to the problem of code equivalence of additive codes over finite extensions of ${\mathbb F}_2$.

Highlights

  • We present a polynomial-time reduction from the multi-graph isomorphism problem to the problem of code equivalence of additive codes over finite extensions of the field with two elements

  • Let F be a finite field of characteristic p, where p is a prime

  • An additive code of length n over F is a subset of F n with the property that for all u, v ∈ C, we have that u + v ∈ C

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Summary

Introduction

Let F be a finite field of characteristic p, where p is a prime. An additive code of length n over F is a subset of F n with the property that for all u, v ∈ C, we have that u + v ∈ C. In [7], Petrank and Roth provide a polynomial-time reduction from the graph isomorphism problem to the binary linear code equivalence problem. Known algorithms for graph isomorphism include McKay’s Nauty algorithm [5], Ullmann’s algorithm [10], the VF2 algorithm [3] and the parameterised matching algorithm [6]. Solving isomorphism generally takes much longer time if there is no match, since all possible mappings are eventually searched until it is shown that there is no isomorphism The latter extends to multigraph isomorphism and is based on a parameterised sequence which is a walk that covers every vertex in the graph. Since a graph on n vertices has at most 21 n(n − 1) edges, we assume that the multi-graph has h 12 n(n − 1) weights and that these are from the set {1, . Since a graph on n vertices has at most 21 n(n − 1) edges, we assume that the multi-graph has h 12 n(n − 1) weights and that these are from the set {1, . . . , h}

The reduction of multi-graph isomorphism to additive code equivalence
Code equivalence algorithms

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