A simple algorithm for decoding Reed-Solomon codes and its relation to the Welch-Berlekamp algorithm

A simple and natural Gao algorithm for decoding algebraic codes is described. Its relation to the Welch-Berlekamp and Euclidean algorithms is given.Comment: 7 pages. Submitted to IEEE Transactions on Information Theor

Long-range correlated random field and random anisotropy O(N) models: A functional renormalization group study

We study the long-distance behavior of the O(N) model in the presence of random fields and random anisotropies correlated as ~1/x^{d-sigma} for large separation x using the functional renormalization group. We compute the fixed points and analyze their regions of stability within a double epsilon=d-4 and sigma expansion. We find that the long-range disorder correlator remains analytic but generates short-range disorder whose correlator develops the usual cusp. This allows us to obtain the phase diagrams in (d,sigma,N) parameter space and compute the critical exponents to first order in epsilon and sigma. We show that the standard renormalization group methods with a finite number of couplings used in previous studies of systems with long-range correlated random fields fail to capture all critical properties. We argue that our results may be relevant to the behavior of He-3A in aerogel.Comment: 8 pages, 3 figures, revtex

Localization of spin waves in disordered quantum rotors

We study the dynamics of excitations in a system of $O(N)$ quantum rotors in the presence of random fields and random anisotropies. Below the lower critical dimension $d_{\mathrm{lc}}=4$ the system exhibits a quasi-long-range order with a power-law decay of correlations. At zero temperature the spin waves are localized at the length scale $L_{\mathrm{loc}}$ beyond which the quantum tunneling is exponentially suppressed $c \sim e^{-(L/L_{\mathrm{loc}})^{2(\theta+1)}}$. At finite temperature $T$ the spin waves propagate by thermal activation over energy barriers that scales as $L^{\theta}$. Above $d_{\mathrm{lc}}$ the system undergoes an order-disorder phase transition with activated dynamics such that the relaxation time grows with the correlation length $\xi$ as $\tau \sim e^{C \xi^\theta/T}$ at finite temperature and as $\tau \sim e^{C' \xi^{2(\theta+1)}/\hbar^2}$ in the vicinity of the quantum critical point.Comment: 8 pages, 2 figures, revtex

Avoiding Echo-Responses in a Retrieval-Based Conversation System

Retrieval-based conversation systems generally tend to highly rank responses that are semantically similar or even identical to the given conversation context. While the system's goal is to find the most appropriate response, rather than the most semantically similar one, this tendency results in low-quality responses. We refer to this challenge as the echoing problem. To mitigate this problem, we utilize a hard negative mining approach at the training stage. The evaluation shows that the resulting model reduces echoing and achieves better results in terms of Average Precision and Recall@N metrics, compared to the models trained without the proposed approach

Instanton theory for bosons in disordered speckle potential

We study the tail of the spectrum for non-interacting bosons in a blue-detuned random speckle potential. Using an instanton approach we derive the asymptotic behavior of the density of states in d dimensions. The leading corrections resulting from fluctuations around the saddle point solution are obtained by means of the Gel'fand-Yaglom method generalized to functional determinants with zero modes. We find a good agreement with the results of numerical simulations in one dimension. The effect of weak repulsive interactions in the Lifshitz tail is also discussed.Comment: 12 pages, 3 figures, revtex

Finding roots of polynomials over finite fields

We propose an improved algorithm for finding roots of polynomials over finite fields. This makes possible significant speedup of the decoding process of Bose-Chaudhuri-Hocquenghem, Reed-Solomon, and some other error-correcting codes.Comment: 6 pages. IEEE Transactions on Communication