Numerical and analytical bounds on threshold error rates for hypergraph-product codes

We study analytically and numerically decoding properties of finite rate hypergraph-product quantum LDPC codes obtained from random (3,4)-regular Gallager codes, with a simple model of independent X and Z errors. Several non-trival lower and upper bounds for the decodable region are constructed analytically by analyzing the properties of the homological difference, equal minus the logarithm of the maximum-likelihood decoding probability for a given syndrome. Numerical results include an upper bound for the decodable region from specific heat calculations in associated Ising models, and a minimum weight decoding threshold of approximately 7%.Comment: 14 pages, 5 figure

Network vector quantization

We present an algorithm for designing locally optimal vector quantizers for general networks. We discuss the algorithm's implementation and compare the performance of the resulting "network vector quantizers" to traditional vector quantizers (VQs) and to rate-distortion (R-D) bounds where available. While some special cases of network codes (e.g., multiresolution (MR) and multiple description (MD) codes) have been studied in the literature, we here present a unifying approach that both includes these existing solutions as special cases and provides solutions to previously unsolved examples

Constant-Weight Gray Codes for Local Rank Modulation

We consider the local rank-modulation scheme in which a sliding window going over a sequence of real-valued variables induces a sequence of permutations. The local rank-modulation, as a generalization of the rank-modulation scheme, has been recently suggested as a way of storing information in flash memory. We study constant-weight Gray codes for the local rank-modulation scheme in order to simulate conventional multi-level flash cells while retaining the benefits of rank modulation. We provide necessary conditions for the existence of cyclic and cyclic optimal Gray codes. We then specifically study codes of weight 2 and upper bound their efficiency, thus proving that there are no such asymptotically-optimal cyclic codes. In contrast, we study codes of weight 3 and efficiently construct codes which are asymptotically-optimal

Constant-Weight Gray Codes for Local Rank Modulation

We consider the local rank-modulation scheme in which a sliding window going over a sequence of real-valued variables induces a sequence of permutations. Local rank- modulation is a generalization of the rank-modulation scheme, which has been recently suggested as a way of storing information in flash memory. We study constant-weight Gray codes for the local rank- modulation scheme in order to simulate conventional multi-level flash cells while retaining the benefits of rank modulation. We provide necessary conditions for the existence of cyclic and cyclic optimal Gray codes. We then specifically study codes of weight 2 and upper bound their efficiency, thus proving that there are no such asymptotically-optimal cyclic codes. In contrast, we study codes of weight 3 and efficiently construct codes which are asymptotically-optimal. We conclude with a construction of codes with asymptotically-optimal rate and weight asymptotically half the length, thus having an asymptotically-optimal charge difference between adjacent cells