On Computing Nearest Neighbors with Applications to Decoding of Binary Linear Codes.

We propose a new decoding algorithm for random binary linear codes. The so-called information set decoding algorithm of Prange (1962) achieves worst-case complexity 2(0.121n). In the late 80s, Stern proposed a sort-and-match version for Prange's algorithm, on which all variants of the currently best known decoding algorithms are build. The fastest algorithm of Becker, Joux, May and Meurer (2012) achieves running time 2(0.102n) in the full distance decoding setting and 2(0.0494n) with half (bounded) distance decoding. In this work we point out that the sort-and-match routine in Stern's algorithm is carried out in a non-optimal way, since the matching is done in a two step manner to realize an approximate matching up to a small number of error coordinates. Our observation is that such an approximate matching can be done by a variant of the so-called High Dimensional Nearest Neighbor Problem. Namely, out of two lists with entries from F-2(m) we have to find a pair with closest Hamming distance. We develop a new algorithm for this problem with sub-quadratic complexity which might be of independent interest in other contexts. Using our algorithm for full distance decoding improves Stern's complexity from 2(0.117n) to 2(0.114n). Since the techniques of Becker et al apply for our algorithm as well, we eventually obtain the fastest decoding algorithm for binary linear codes with complexity 2(0.097n). In the half distance decoding scenario, we obtain a complexity of 2(0.0473n).