Statistical mechanics of error-correcting codes

We investigate the performance of error-correcting codes, where the code word comprises products of K bits selected from the original message and decoding is carried out utilizing a connectivity tensor with C connections per index. Shannon's bound for the channel capacity is recovered for large K and zero temperature when the code rate K/C is finite. Close to optimal error-correcting capability is obtained for finite K and C. We examine the finite-temperature case to assess the use of simulated annealing for decoding and extend the analysis to accommodate other types of noisy channels.

Simple Gallager codes for BSC

We investigate the performance of Gallager type error- correcting codes for Binary Symmetric Channels, where the code word comprises products of K bits selected from the original message and decoding is carried out utilizing a connectivity tensor with C connections per index. Shannon's bound for the channel capacity is recovered for large K and zero temperature when the code rate K/C is finite. Close to optimal error-correcting capability, with improved decoding properties is obtained for finite K and C.

Statistical mechanics of mutual information maximization

An unsupervised learning procedure based on maximizing the mutual information between the outputs of two networks receiving different but statistically dependent inputs is analyzed (Becker S. and Hinton G., Nature, 355 (1992) 161). By exploiting a formal analogy to supervised learning in parity machines, the theory of zero-temperature Gibbs learning for the unsupervised procedure is presented for the case that the networks are perceptrons and for the case of fully connected committees.

Finite-connectivity spin-glass phase diagrams and low-density parity check codes

We obtain phase diagrams of regular and irregular finite-connectivity spin glasses. Contact is first established between properties of the phase diagram and the performance of low-density parity check (LDPC) codes within the replica symmetric (RS) ansatz. We then study the location of the dynamical and critical transition points of these systems within the one step replica symmetry breaking theory (RSB), extending similar calculations that have been performed in the past for the Bethe spin-glass problem. We observe that the location of the dynamical transition line does change within the RSB theory, in comparison with the results obtained in the RS case. For LDPC decoding of messages transmitted over the binary erasure channel we find, at zero temperature and rate R=14, an RS critical transition point at pc 0.67 while the critical RSB transition point is located at pc 0.7450±0.0050, to be compared with the corresponding Shannon bound 1-R. For the binary symmetric channel we show that the low temperature reentrant behavior of the dynamical transition line, observed within the RS ansatz, changes its location when the RSB ansatz is employed; the dynamical transition point occurs at higher values of the channel noise. Possible practical implications to improve the performance of the state-of-the-art error correcting codes are discussed. © 2006 The American Physical Society.

Crossover evaluation of silicone hydrogel daily disposable contact lenses

PURPOSE: To assess the surface tear breakup time and clinical performance of three daily disposable silicone hydrogel contact lenses over 16 hours of wear. METHODS: Thirty-nine patients (mean [±SD] age, 22.1 [±3.5] years) bilaterally wore (narafilcon A, filcon II-3, and delefilcon A) contact lenses in a prospective, randomized, masked, 1-week crossover clinical trial. Tear film was assessed by the tear meniscus height (TMH), ocular/contact lens surface temperature dynamics, and lens surface noninvasive breakup time at 8, 12, and 16 hours of wear. Clinical performance and ocular physiology were assessed by subjective questionnaire, by high-/low-contrast logMAR (logarithm of the minimum angle of resolution) acuity, and through bulbar and limbal hyperemia grading. Corneal and conjunctival staining were assessed after lens removal. RESULTS: Delefilcon A demonstrated a longer noninvasive breakup time (13.4 [±4.4] seconds) than filcon II-3 (11.6 [±3.7] seconds; p < 0.001) and narafilcon A (12.3 [±3.7] seconds; p < 0.001). A greater TMH (0.35 [±0.11] mm) was shown by delefilcon A than filcon II-3 (0.32 [±0.10] seconds; p = 0.016). Delefilcon A showed less corneal staining after 16 hours of lens wear (0.7 [±0.6] Efron grade) than filcon II-3 (1.1 [±0.7]; p < 0.001) and narafilcon A (0.9 [±0.7]; p = 0.031). Time was not a significant factor for prelens tear film stability (F = 0.594, p = 0.555) or TMH (F = 0.632, p = 0.534). Lens brand did not affect temperature (F = 1.220, p = 0.308), but it decreased toward the end of the day (F = 19.497, p < 0.001). Comfort, quality of vision, visual acuity and contrast acuity, and limbal grading were similar between the lens brands but decreased with time during the day (p < 0.05). CONCLUSIONS: The tear breakup time over the contact lens surface differed between lens types and may have a role in protecting the ocular surface.

Kelvin-wave cascade in the vortex filament model

The small-scale energy-transfer mechanism in zero-temperature superfluid turbulence of helium-4 is still a widely debated topic. Currently, the main hypothesis is that weakly nonlinear interacting Kelvin waves (KWs) transfer energy to sufficiently small scales such that energy is dissipated as heat via phonon excitations. Theoretically, there are at least two proposed theories for Kelvin-wave interactions. We perform the most comprehensive numerical simulation of weakly nonlinear interacting KWs to date and show, using a specially designed numerical algorithm incorporating the full Biot-Savart equation, that our results are consistent with the nonlocal six-wave KW interactions as proposed by L'vov and Nazarenko.