Lattice (List) Decoding Near Minkowski's Inequality
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Minkowski proved that any n-dimensional lattice of unit determinant has a nonzero vector of Euclidean norm at most √(n); in fact, there are 2^Ω(n) such lattice vectors. Lattices whose minimum distances come close to Minkowski's bound provide excellent sphere packings and error-correcting codes in ℝ^n. The focus of this work is a certain family of efficiently constructible n-dimensional lattices due to Barnes and Sloane, whose minimum distances are within an O(√(log n)) factor of Minkowski's bound. Our primary contribution is a polynomial-time algorithm that list decodes this family to distances approaching 1/√(2) times the minimum distance. The main technique is to decode Reed-Solomon codes under error measured in the Euclidean norm, using the Koetter-Vardy "soft decision" variant of the Guruswami-Sudan list-decoding algorithm.
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