PCP-free APX-Hardness of Nearest Codeword and Minimum Distance
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| Format: | Preprint |
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2025
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| _version_ | 1866915357975904256 |
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| author | Bhattiprolu, Vijay Guruswami, Venkatesan Ren, Xuandi |
| author_facet | Bhattiprolu, Vijay Guruswami, Venkatesan Ren, Xuandi |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_11131 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | PCP-free APX-Hardness of Nearest Codeword and Minimum Distance Bhattiprolu, Vijay Guruswami, Venkatesan Ren, Xuandi Computational Complexity 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. |
| title | PCP-free APX-Hardness of Nearest Codeword and Minimum Distance |
| topic | Computational Complexity |
| url | https://arxiv.org/abs/2503.11131 |