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Register a new userOn the Correlation Distribution for a Ternary Niho Decimation
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Nian Li
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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In this paper, let $n=2m$ and $d=3^{m+1}-2$ with $mgeq2$ and $gcd(d,3^n-1)=1$. By studying the weight distribution of the ternary Zetterberg code and counting the numbers of solutions of some equations over the finite field $mathbb{F}_{3^n}$, the correlation distribution between a ternary $m$-sequence of period $3^n-1$ and its $d$-decimation sequence is completely determined. This is the first time that the correlation distribution for a non-binary Niho decimation has been determined since 1976.
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We introduce a two-stage decimation process to improve the performance of neural belief propagation (NBP), recently introduced by Nachmani et al., for short low-density parity-check (LDPC) codes. In the first stage, we build a list by iterating between a conventional NBP decoder and guessing the least reliable bit. The second stage iterates between a conventional NBP decoder and learned decimation, where we use a neural network to decide the decimation value for each bit. For a (128,64) LDPC code, the proposed NBP with decimation outperforms NBP decoding by 0.75 dB and performs within 1 dB from maximum-likelihood decoding at a block error rate of $10^{-4}$.
Motivated by recent results on the constructions of permutation polynomials with few terms over the finite field $mathbb{F}_{2^n}$, where $n$ is a positive even integer, we focus on the construction of permutation trinomials over $mathbb{F}_{2^n}$ from Niho exponents. As a consequence, several new classes of permutation trinomials over $mathbb{F}_{2^n}$ are constructed from Niho exponents based on some subtle manipulation of solving equations with low degrees over finite fields.
While iterative quantizers based on low-density generator-matrix (LDGM) codes have been shown to be able to achieve near-ideal distortion performance with comparatively moderate block length and computational complexity requirements, their analysis remains difficult due to the presence of decimation steps. In this paper, considering the use of LDGM-based quantizers in a class of symmetric source coding problems, with the alphabet being either binary or non-binary, it is proved rigorously that, as long as the degree distribution satisfies certain conditions that can be evaluated with density evolution (DE), the belief propagation (BP) marginals used in the decimation step have vanishing mean-square error compared to the exact marginals when the block length and iteration count goes to infinity, which potentially allows near-ideal distortion performances to be achieved. This provides a sound theoretical basis for the degree distribution optimization methods previously proposed in the literature and already found to be effective in practice.
We investigate an encoding scheme for lossy compression of a binary symmetric source based on simple spatially coupled Low-Density Generator-Matrix codes. The degree of the check nodes is regular and the one of code-bits is Poisson distributed with an average depending on the compression rate. The performance of a low complexity Belief Propagation Guided Decimation algorithm is excellent. The algorithmic rate-distortion curve approaches the optimal curve of the ensemble as the width of the coupling window grows. Moreover, as the check degree grows both curves approach the ultimate Shannon rate-distortion limit. The Belief Propagation Guided Decimation encoder is based on the posterior measure of a binary symmetric test-channel. This measure can be interpreted as a random Gibbs measure at a temperature directly related to the noise level of the test-channel. We investigate the links between the algorithmic performance of the Belief Propagation Guided Decimation encoder and the phase diagram of this Gibbs measure. The phase diagram is investigated thanks to the cavity method of spin glass theory which predicts a number of phase transition thresholds. In particular the dynamical and condensation phase transition temperatures (equivalently test-channel noise thresholds) are computed. We observe that: (i) the dynamical temperature of the spatially coupled construction saturates towards the condensation temperature; (ii) for large degrees the condensation temperature approaches the temperature (i.e. noise level) related to the information theoretic Shannon test-channel noise parameter of rate-distortion theory. This provides heuristic insight into the excellent performance of the Belief Propagation Guided Decimation algorithm. The paper contains an introduction to the cavity method.
In this paper, a class of permutation trinomials of Niho type over finite fields with even characteristic is further investigated. New permutation trinomials from Niho exponents are obtained from linear fractional polynomials over finite fields, and it is shown that the presented results are the generalizations of some earlier works.
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