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The Goldbach conjecture states that every even integer ≥ 4 can be written as a sum of two prime numbers. It is known to be true up to 4 × 1011. In this paper, new experiments on a Cray C916 supercomputer and on an SGI compute server with 18 R10000 CPUs are described, which extend this bound to 1014. Two consequences are that (1) under the assumption of the Generalized Riemann hypothesis, every odd...
Permutation design is a crucial point for turbocodes in terms of error correction performance. Many permutation design models have been proposed to deal with this point. However, most of them are defined in terms of several free parameters that have to be correctly chosen. The selection of these parameters generally leads to a systematic search in a wide domain of parameters value. In this article,...
Thanks to the probabilistic message passing performed between its component decoders, a turbo decoder is able to provide strong error correction close to the theoretical limit. However, the minimum Hamming distance (dmin) of a turbo code may not be sufficiently large to ensure large asymptotic gains at very low error rates (the so-called flattening effect). Increasing the dmin of a turbo code may...
Permutation design is a crucial point for turbocodes in terms of error correction performance. Many permutation design models have been proposed to deal with this point. However, most of them are defined in terms of several free parameters that have to be correctly chosen. In this article, we describe a new criterion to optimize the choice of the permutation. Then we use it with two other standard...
In this paper, we describe a new method to determine the Frame Error Rate (FER) for error-correcting codes in the presence of additive white Gaussian noise (AGWN) under a maximum-likelihood (ML) hypothesis. This algorithm is a generalization of the impulse method of Berrou et al. We compare our method with the standard Monte Carlo method for the estimate of low frame error rates for the extended Golay...
Thanks to the message passing principle, turbo decoding is able to provide strong error correction near the theoretical (Shannon) limit. However, the minimum Hamming distance (MHD) of a turbo code may not be sufficient to prevent a detrimental change in the error rate vs. signal to noise ratio curve, the so-called flattening. Increasing the MHD of a turbo code may involve using component encoders...
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