Hardness of Euclidean Closest Vector within n1/8−ϵ and Binary Nearest Codeword within n1/4−ϵ

Mikhail Alekhnovich, Subhash Khot, Guy Kindler +1 more

Zhao Song

We prove two deterministic inapproximability results. First, for every fixed $0<\epsilon<1/2$, Euclidean $\mathrm{GapCVP}^{(2)}$ is NP-hard with gap factor $n^{1/2-\epsilon}$ under deterministic polynomial-time many-one reductions, where $n$ denotes the lattice rank. Consequently, the Euclidean closest vector problem is NP-hard to approximate within the same factor. This improves the previous $n^{1/400}$ hardness factor in Chapter 7 of the OpenAI report [Ope25]. Aharonov and Regev gave short certificates for both the YES and the NO case of $\operatorname{GapCVP}^{(2)}$ at gap factor \( C\sqrt n \), placing that problem in $\mathrm{NP}\cap\mathrm{coNP}$ for an absolute constant $C>0$ [AR05]. An NP-hard problem lying in $\mathrm{coNP}$ would give $\mathrm{NP}=\mathrm{coNP}$, so the factor $n^{1/2-\epsilon}$ above cannot be improved to \( C\sqrt n \) unless the two classes coincide. Second, for every fixed $0<\epsilon<1$, the gap versions of binary nearest codeword and binary syndrome decoding are NP-hard with factor $n^{1-\epsilon}$ under deterministic polynomial-time many-one reductions, where $n$ denotes the binary block length. Consequently, both optimization problems are NP-hard to approximate within the same factor. This improves the previous $n^{1/200}$ hardness factor in Chapter 7 of the OpenAI report [Ope26].

Chandan K. Dubey, Thomas Holenstein

Saïd Nouh, Achraf El Khatabi, Mostafa Belkasmi

In this paper we present an efficient algorithm to decode linear block codes on binary channels. The main idea consists in using a vote procedure in order to elaborate artificial reliabilities of the binary received word and to present the obtained real vector r as inputs of a SIHO decoder (Soft In/Hard Out). The goal of the latter is to try to find the closest codeword to r in terms of the Euclidean distance. A comparison of the proposed algorithm over the AWGN channel with the Majority logic decoder, Berlekamp-Massey, Bit Flipping, Hartman-Rudolf algorithms and others show that it is more efficient in terms of performance. The complexity of the proposed decoder depends on the weight of the error to decode, on the code structure and also on the used SIHO decoder.

Baolong Jin, Rui Xue

J. Conway, Neil J.A. Sloane

Two kinds of algorithms are considered.1)If *** is a binary code of lengthn, a "soft decision" decoding algorithm for *** changes an arbitrary point ofR^{n}into a nearest codeword (nearest in Euclidean distance).2)Similarly, a decoding algorithm for a lattice\LambdainR^{n}changes an arbitrary point ofR^{n}into a closest lattice point. Some general methods are given for constructing such algorithms, ami are used to obtain new and faster decoding algorithms for the Gosset latticeE_{8}, the Golay code the Leech lattice.