The aim of this paper is to prove the achievability of several coding problems by using sparse matrices (the maximum column weight grows logarithmically in the block length) and maximal-likelihood (ML) coding. These problems are the Slepian-Wolf problem, the Gel'fand-Pinsker problem, the Wyner-Ziv problem, and the Onehelps-one problem (source coding with partial side information at the decoder). To this end, the notion of a hash property for an ensemble of functions is introduced and it is proved that an ensemble of q-ary sparse matrices satisfies the hash property. Based on this property, it is proved that the rate of codes using sparse matrices and maximal-likelihood (ML) coding can achieve the optimal rate.
Index TermsShannon theory, hash functions, linear codes, sparse matrix, maximum-likelihood eoncoding/decoding, the Slepian-Wolf problem, the Gel'fand-Pinsker problem, the Wyner-Ziv problem, the One-helps-one problem
I. INTRODUCTIONThe aim of this paper is to prove the achievability of several coding problems by using sparse matrices (the maximum column weight grows logarithmically in the block length) and maximal-likelihood (ML) coding 1 , namely the Slepian-Wolf problem [39] (Fig. 1), the Gel'fand-Pinsker problem [13] (Fig. 2), the Wyner-Ziv problem [47] (Fig. 3), and the One-helps-one problem (source coding with partial side information at the decoder) [44][46] (Fig. 4). To prove these theorems, we first introduce the notion of a hash property for an ensemble of functions, where functions are not assumed to be linear. This notion is a sufficient condition for the achievability of coding theorems. Next, we prove that an ensemble of q-ary sparse matrices, which is an extension of [21], satisfies the hash property. Finally, based on the hash property, we prove that the rate of J. Muramatsu is with NTT
Stochastic encoders for channel coding and lossy source coding are introduced with a rate close to the fundamental limits, where the only restriction is that the channel input alphabet and the reproduction alphabet of the lossy source code are finite. Random numbers, which satisfy a condition specified by a function and its value, are used to construct stochastic encoders. The proof of the theorems is based on the hash property of an ensemble of functions, where the results are extended to general channels/sources and alternative formulas are introduced for channel capacity and the rate-distortion region. Since an ensemble of sparse matrices has a hash property, we can construct a code by using sparse matrices, where the sum-product algorithm can be used for encoding and decoding by assuming that channels/sources are memoryless.
We propose a secure key distribution scheme based on correlated physical randomness in remote optical scramblers driven by common random light. The security of the scheme depends on the practical difficulty of completely observing random optical phenomena. We describe a particular realization using the synchronization of semiconductor lasers injected with common light of randomly varying phase. We experimentally demonstrate the feasibility of the scheme over a distance of 120 km.
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