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Tail Behavior of Sphere-Decoding Complexity in Random Lattices

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Dominik Seethaler, Joakim Jaldén, Christoph Studer, Helmut Bölcskei

Information Theory · Submitted 8 May 2009

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In particular, we show that under fairly general assumptions on the statistics of the lattice basis matrix, the tail behavior of the SD complexity distribution is solely determined by the inverse volume of a fundamental region of the underlying lattice.

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arXiv ID0905.1215 (version 1)
Pith IDHCUX7OT7, Pith's permanent record of this paper and its review
CitationChristoph Studer, Dominik Seethaler, Helmut B\"olcskei, Joakim Jald\'en. "Tail Behavior of Sphere-Decoding Complexity in Random Lattices." arXiv:0905.1215 (2009). https://pith.science/paper/HCUX7OT7
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,222 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.48 · human participants 0.01 · controlled comparison 0.16 · randomized assignment 0.02 · states quantitative result 0.97 · compares with prior results 0.67 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We analyze the (computational) complexity distribution of sphere-decoding (SD) for random infinite lattices. In particular, we show that under fairly general assumptions on the statistics of the lattice basis matrix, the tail behavior of the SD complexity distribution is solely determined by the inverse volume of a fundamental region of the underlying lattice. Particularizing this result to NxM, N>=M, i.i.d. Gaussian lattice basis matrices, we find that the corresponding complexity distribution is of Pareto-type with tail exponent given by N-M+1. We furthermore show that this tail exponent is not improved by lattice-reduction, which includes layer-sorting as a special case.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We analyze the (computational) complexity distribution of sphere-decoding (SD) for random infinite lattices. contribution 0.99 0.93 0.06 asserted 0.88
s1 In particular, we show that under fairly general assumptions on the statistics of the lattice basis matrix, the tail behavior of the SD complexity distribution is solely determined by the inverse volume of a fundamental region of the underlying lattice. result 0.99 0.94 0.81 asserted 1.00
s2 Particularizing this result to NxM, N>=M, i.i.d. contribution 0.57 0.62 0.04 unclear 0.91
s3 Gaussian lattice basis matrices, we find that the corresponding complexity distribution is of Pareto-type with tail exponent given by N-M+1. result 1.00 0.95 0.20 asserted 1.00
s4 We furthermore show that this tail exponent is not improved by lattice-reduction, which includes layer-sorting as a special case. result 1.00 0.93 0.18 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We analyze the (computational) complexity distribution of sphere-decoding (SD) for random infinite lattices. In particular, we show that under fairly general assumptions on the statistics of the lattice basis matrix, the tail behavior of the SD complexity distribution is solely determined by the inverse volume of a fundamental region of the underlying lattice. Particularizing this result to NxM, N>=M, i.i.d. Gaussian lattice basis matrices, we find that the corresponding complexity distribution is of Pareto-type with tail exponent given by N-M+1. We furthermore show that this tail exponent is not improved by lattice-reduction, which includes layer-sorting as a special case.

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