Hardness of approximating the weight enumerator of a binary linear code

TL;DR

This paper proves that exactly computing or approximating the weight enumerator of a binary linear code is computationally hard, even for approximate evaluations at specific points, indicating significant complexity in coding theory problems.

Contribution

It establishes the computational hardness of both exact and approximate evaluation of the weight enumerator, including at specific complex points, within the polynomial hierarchy.

Findings

Exact evaluation is hard for polynomial hierarchy.

Approximate evaluation with additive accuracy is also hard.

Evaluation at a specific complex point is hard for certain approximation parameters.

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 $2^{\alpha n}$, $\alpha$ is constant the problem is hard in the above sense. We also prove that approximate evaluation at a single point $e^{\pi i/4}$ is hard for $0<\al<\al_0\approx0.88$.