Finding the permutation between equivalent linear codes: the support splitting algorithm

Two linear codes are permutation-equivalent if they are equal up to a fixed permutation on the codeword coordinates. We present here an algorithm able to compute this permutation. It operates by determining a set of properties invariant by permutation, one for each coordinate, called a signature. If this signature is fully discriminant-i.e., different for all coordinates-the support of the code splits into singletons, and the same signature computed for any permutation-equivalent code will allow the reconstruction of the permutation. A procedure is described to obtain a fully discriminant signature for most linear codes. The total complexity of the support splitting algorithm is polynomial in the length of the code and exponential in the dimension of its hull, i.e., the intersection of the code with its dual.