Hardness of approximating the weight enumerator of a binary linear code

Abstract

We consider the problem of evaluation of the weight enumerator of a binary linear code. We show that the exact evaluation is hard for polynomial hierarchy. More exactly, if WE is an oracle answering the solution of the evaluation problem then P^WE=P^GapP. Also we consider the approximative evaluation of the weight enumerator. In the case of approximation with additive accuracy $2^{αn}$, $α$ is constant the problem is hard in the above sense. We also prove that approximate evaluation at a single point $e^{πi/4}$ is hard for $0<\al<\al_0\approx0.88$.