We give a simpliied proof of a theorem of Lagarias, Lenstra and Schnorr 17] that the problem of approximating the length of the shortest lattice vector within a factor of Cn, for an appropriate constant C, cannot be NP-hard, unless NP = coNP. We also prove that the problem of ndng a n 1=4-unique shortest lattice vector is not NP-hard under polynomial time many-one reductions, unless the polynom...

Lattice vector quantization (LVQ) reduces computational load and design complexity due to its regular structure. In this letter, we introduce and analyze the performance of two hybrid combinations of two lattices i.e. the AnAn and AnDn. Experiment results show that multistage LVQ with lattice AnA combination in four dimensional vector offers the least quantization errors with p = 0.0098 as comp...

In this paper, we focus on the structured multi-frame vectors in Hilbert $C^*$-modules. More precisely, it will be shown that the set of all complete multi-frame vectors for a unitary system can be parameterized by the set of all surjective operators, in the local commutant. Similar results hold for the set of all complete wandering vectors and complete multi-Riesz vectors, when the surjective ...

In this paper we make a first feasibility analysis for implementing lattice reduction algorithms on GPU using CUDA, a programming framework for NVIDIA graphics cards. The enumeration phase of the BKZ lattice reduction algorithm is chosen as a good candidate for massive parallelization on GPU. Given the nature of the problem we gain large speedups compared to previous CPU implementations. Our im...

From the above definition, we have the following geometrical interpretation of the dual lattice. For any vector x, the set of all points whose inner product with x is integer forms a set of hyperplanes perpendicular to x and separated by distance 1/‖x‖. Hence, any vector x in a lattice Λ imposes the constraint that all points in Λ∗ lie in one of the hyperplanes defined by x. See the next figure...

We prove that given a point ~z outside a given lattice L then there is a dual vector which gives a fairly good estimate for how far from the lattice the vector is. To be more precise, there is a set of translated hyperplanes Hi such that L S iHi and d(~z; S iHi) 1 6n+1 d(~z; L). Warning: Essentially this paper has been published in Combinatorica and is hence subject to copyright restrictions. I...

Let n ≥ 3. From the description of subdirectly irreducible complemented Arguesian lattices with four generators given by Herrmann, Ringel and Wille it follows that the subspace lattice of an n-dimensional vector space over a finite field is generated by four elements if and only if the field is a prime field. By exhibiting a 5-element generating set we prove that the subspace lattice of an n-di...

Quenched and partially quenched chiral perturbation theory for vector mesons is developed and is used to extract chiral loop correction to the ρ meson mass. Connections to fully quenched and totally unquenched chiral perturbation theory results are discussed. It is also shown that (partially) quenched perturbation theory for tensor mesons can be formulated analogously, and the chiral correction...

This paper proposed an even grid based lattice vector quantization method for audio coding. The method uses energy priority, with basic code book and the ball-type expansion, which is applicable to the low rate of the variable-rate vector quantization coding. The method uses the lattice characteristics to resolve rapid index distribution problem, as well as the compression of the basic code boo...

We elucidate the properties of mixed-gap vector surface solitons supported by the interface between a uniform medium and an optical lattice imprinted in a Kerr-type nonlinear medium. The components of such mixed-gap solitons emerge from different gaps of lattice spectrum and their mutual trapping results in the formation of stable vector states. The unstable soliton component is stabilized by t...