PCP-free APX-Hardness of Nearest Codeword and Minimum Distance

Vijay Bhattiprolu; Venkatesan Guruswami; Xuandi Ren

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arXiv · 2503.11131

PCP-free APX-Hardness of Nearest Codeword and Minimum Distance

Vijay Bhattiprolu·Venkatesan Guruswami·Xuandi Ren

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Abstract

We give simple deterministic reductions demonstrating the NP-hardness of approximating the nearest codeword problem and minimum distance problem within arbitrary constant factors (and almost-polynomial factors assuming NP cannot be solved in quasipolynomial time). The starting point is a simple NP-hardness result without a gap, and is thus "PCP-free." Our approach is inspired by that of Bhattiprolu and Lee [BL24] who give a PCP-free randomized reduction for similar problems over the integers and the reals. We leverage the existence of $\varepsilon$-balanced codes to derandomize and further simplify their reduction for the case of finite fields.

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Vijay Bhattiprolu, Venkatesan Guruswami, Xuandi Ren. 2025-03-14. PCP-free APX-Hardness of Nearest Codeword and Minimum Distance. https://arxiv.org/abs/2503.11131

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