Locally Dense Codes

Abstract

The Minimum Distance Problem (MDP), i.e., the computational task of evaluating (exactly or approximately) the minimum distance of a linear code, is a well known NP-hard problem in coding theory. A key element in essentially all known proofs that MDP is NP-hard is the construction of a combinatorial object that we may call a locally dense code. This is a linear code with large minimum distance d that admits a ball of smaller radius r¡d containing an exponential number of codewords, together with some auxiliary information used to map these codewords. In this paper we provide a generic method to explicitly construct locally dense binary codes, starting from an arbitrary linear code with sufficiently large minimum distance. Instantiating our construction with well known linear codes (e.g., Reed-Solomon codes concatenated with Hadamard codes) yields a simple proof that MDP is NPhard to approximate within any constant factor under deterministic polynomial time reductions, simplifying and explaining recent results of Cheng and Wan (STOC 2009 / IEEE Trans. Inf. Theory, 2012) and Austrin and Khot (ICALP 2011). Our work is motivated by the construction of analogous combinatorial objects over integer lattices, which are used in NP-hardness proofs for the Shortest Vector Problem (SVP). We show that for the max norm, locally dense lattices can also be easily constructed. However, all currently known constructions of locally dense lattices in the standard Euclidean norm are probabilistic. Finding a deterministic construction of locally dense Euclidean lattices, analogous to the results presented in this paper, would prove the NP-hardness of approximating SVP under deterministic polynomial time reductions, a long standing open problem in the computational complexity of integer lattices.