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

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Coherent growth and mechanical properties of AlN/VN multilayers J. Appl. Phys. 95, 92 (2004); 10.1063/1.1630367Synthesis and mechanical properties of boron suboxide thin films By controlling the time period for which the substrate was positioned opposite each of the graphite and boron nitride semicircular targets, nanometer period carbon nitride and boron nitride multilayer films were deposited one after another in an argon and nitrogen atmosphere. Both the nanoindentation hardness and microwear resistance of the multilayer films changed with the layer period. The multilayer film with a 4 nm period had the highest hardness and microwear resistance. To clarify the reason for this, the atomic scale wear was characterized. The rate of increase of wear depth was almost zero at the depth which corresponds to each layer thickness. It is suggested that the interfaces prevent the defect elongation. The preventive effect of the interface on defect elongation of a 4 nm period multilayer film is greater than that of a 2 nm period multilayer film.

Atomic-scale mechanical processing of layered materials such as muscovite mica was performed using an atomic force microscope (AFM). Processing began at a certain critical load above 130 nN, and the processing depth increased discretely with load. Fracture easily occurred at the two cleavage planes of SiO4–K and K–SiO4 interfaces. With a load slightly larger than the critical load, several repetitions of mechanical sliding of the tip generated a 1 nm deep groove which corresponds to the distance from the top surface of SiO4 to the top surface of the next SiO4 layer beneath it with the removal of residual potassium on the surface. For example, a groove with four steps of 1 nm depth was processed by step-by-step mechanical sliding.

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