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For the additive white Gaussian noise channel, there is a gap between the channel capacity and the highest achievable rate of equiprobable uniformly spaced one-dimensional signaling. It is commonly believed that approaching channel capacity requires the constituent one-dimensional signal points to have a Gaussian probability distribution. It is shown that the channel capacity can be achieved by equiprobable signaling with geometrical, Gaussian-like signal sets. Construction of these signal constellations is explicitly given. This result implicates that it is possible to approach channel capacity without using any kind of shaping codes. >

New multilevel constructions of the Leech lattice and the Golay code are presented. They are derived from Turyn's constructions and the 'holy construction' with the octacode as the glue code. Further, we show that the 'holy construction' of the Leech lattice with the octacode as the glue code is essentially different from the permuted Turyn construction, although both constructions rely on the octacode. Based on these structures, more efficient bounded-distance decoding algorithms of the Golay code and the Leech lattice are presented. >

Several new algorithms for decoding lattice partitions are presented. They apply to Viterbi decoding of multidimensional trellis codes based on these partitions. In [1, 2], trellisbased algorithms were presented for decoding the lattice partitions. The new algorithms can achieve about 50% reduction of the complexity of decoding the lattice partitions in terms of real additions/comparisons compared with the algorithms of [1, 2]. The complexity of the resulting overall Viterbi decoding algorithms still shows a modest improvement. An algorithm for soft decision decoding the first-order Reed-Muller code (8, 4, 4) or the Gosset lattice is also presented. It involves at most 17 real operations, thus, improving the best known algorithm.

Binary block codes for correcting asymmetric errors are called binary AsEC block codes. With the properties of perfect codes for the binary symmetric channel in mind, natural definitions of perfect, weakly perfect and uniformly weakly perfect binary AsEC block codes are given and their properties are studied.