  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.
