Sphere Packings, Lattices and Groups

Conway, John H.; Sloane, N. J. A.

doi:10.1007/978-1-4757-2016-7

Cited by 1,108 publications

(462 citation statements)

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“…The F 4 ∧ F 5 ∧ F 4 structure of our effective action is intriguingly reminiscent of the construction of the Golay code from three 8-bit Hamming codes [24] (or equivalently the Leech lattice from three E 8 root lattices [25]). It is also interesting to note in this connection that our ansatz (4) for the 4-forms F 4 , which involved extending an SO(3) gauge connection in 3+1 dimensions to an SO(8) connection in 7+1 dimensions, has an intimate connection with octonians, which of course provides a connection with the E 8 root lattice.…”

Section: Resultsmentioning

confidence: 97%

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A Unique Theory of Gravity and Matter

Chapline

1999

Chaos, Solitons & Fractals

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The author has previously suggested that the ground state for 4-dimensional quantum gravity can be represented as a condensation of nonlinear gravitons connected by Dirac strings. In this note we suggest that the low-lying excitations of this state can be described by a quasitopological action of the form d z 13 ∫ F 4 ∧ F 5 ∧ F 4 , corresponding to a trilinear coupling of solotonic 8-branes and 7-branes. It is shown that when the excitations associated with F 5 are neglected, the effective action can be interpreted as a theory of conformal gravity in four dimensions. This in turn suggests that ordinary gravity as well supersymmetric matter and phenomenological gauge symmetries arise from the spontaneous breaking of topological invariance. The possibly deep mathematical significance of this theory is also noted.

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Section: Resultsmentioning

confidence: 97%

“…As is well known to specialists in group theory there is a close connection between the periodic arrangements of atoms in crystalline solids and the root diagrams for certain Lie algebras [24]. For example the root diagram for SU(4) can be identified with the unit cell for a facecentered cubic lattice.…”

Section: Gauge Symmetries As Crystallographic Symmetriesmentioning

confidence: 95%

A Unique Theory of Gravity and Matter

Chapline

1999

Chaos, Solitons & Fractals

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“…We refer to [18] for a better numerical upper bound and to [48,19] for a lower and an upper asymptotical bounds.…”

Section: Theorem 2 (Minkowski [49]) Letmentioning

confidence: 99%

Recursive Lattice Reduction

Plantard

Susilo

2010

Lecture Notes in Computer Science

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Abstract. Lattice reduction is known to be a very powerful tool in modern cryptanalysis. In the literature, there are many lattice reduction algorithms that have been proposed with various time complexity (from quadratic to subexponential). These algorithms can be utilized to find a short vector of a lattice with a small norm. Over time, shorter vector will be found by incorporating these methods. In this paper, we take a different approach by presenting a methodology that can be applied to any lattice reduction algorithms, with the implication that enables us to find a shorter vector (i.e. a smaller solution) while requiring shorter computation time. Instead of applying a lattice reduction algorithm to a complete lattice, we work on a sublattice with a smaller dimension chosen in the function of the lattice reduction algorithm that is being used. This way, the lattice reduction algorithm will be fully utilized and hence, it will produce a better solution. Furthermore, as the dimension of the lattice becomes smaller, the time complexity will be better. Hence, our methodology provides us with a new direction to build a lattice that is resistant to lattice reduction attacks. Moreover, based on this methodology, we also propose a recursive method for producing an optimal approach for lattice reduction with optimal computational time, regardless of the lattice reduction algorithm used. We evaluate our technique by applying it to break the lattice challenge 1 by producing the shortest vector known so far. Our results outperform the existing known results and hence, our results achieve the record in the lattice challenge problem.

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“…According to Lemma 9, we have (20) where is defined by (11) and is the noisy received codeword. Therefore, for , the lattice points whose consistent Gaussians will have largest product-of-amplitudes is the point for which is minimized.…”

Section: Convergence Of the Amplitudesmentioning

confidence: 99%

“…However, a reasonable conjecture is to assume that that maximizes the steady-state excitation will also maximize the term that depends on the transient behavior. This means that a reasonable conjecture is to assume that the "winning" lattice point for will also minimize an expression of the form (20).…”

Section: Formentioning

confidence: 99%

Low Density Lattice Codes

Sommer

Feder

Shalvi

2006

2006 IEEE International Symposium on Information Theory

135

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Abstract-Low-density lattice codes (LDLC) are novel lattice codes that can be decoded efficiently and approach the capacity of the additive white Gaussian noise (AWGN) channel. In LDLC a codeword x is generated directly at the n-dimensional Euclidean space as a linear transformation of a corresponding integer message vector b, i.e., x = G G Gb, where H H H = G G G 01 is restricted to be sparse. The fact that H H H is sparse is utilized to develop a linear-time iterative decoding scheme which attains, as demonstrated by simulations, good error performance within 0.5 dB from capacity at block length of n =100,000 symbols. The paper also discusses convergence results and implementation considerations.

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Sphere Packings, Lattices and Groups

Cited by 1,108 publications

References 0 publications

A Unique Theory of Gravity and Matter

A Unique Theory of Gravity and Matter

Recursive Lattice Reduction

Low Density Lattice Codes

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