Asymptotic Distribution of Multilevel Channel Polarization for a Certain Class of Erasure Channels

This study examines multilevel channel polarization for a certain class of erasure channels that the input alphabet size is an arbitrary composite number. We derive limiting proportions of partially noiseless channels for such a class. The results of this study are proved by an argument of convergent sequences, inspired by Alsan and Telatar's simple proof of polarization, and without martingale convergence theorems for polarization process.Comment: 31 pages; 1 figure; 1 table; a short version of this paper has been submitted to the 2018 IEEE International Symposium on Information Theory (ISIT2018

Countably Infinite Multilevel Source Polarization for Non-Stationary Erasure Distributions

Polar transforms are central operations in the study of polar codes. This paper examines polar transforms for non-stationary memoryless sources on possibly infinite source alphabets. This is the first attempt of source polarization analysis over infinite alphabets. The source alphabet is defined to be a Polish group, and we handle the Ar{\i}kan-style two-by-two polar transform based on the group. Defining erasure distributions based on the normal subgroup structure, we give recursive formulas of the polar transform for our proposed erasure distributions. As a result, the recursive formulas lead to concrete examples of multilevel source polarization with countably infinite levels when the group is locally cyclic. We derive this result via elementary techniques in lattice theory.Comment: 12 pages, 1 figure, a short version has been accepted by the 2019 IEEE International Symposium on Information Theory (ISIT2019

Optimality of Huffman Code in the Class of 1-bit Delay Decodable Codes

For a given independent and identically distributed (i.i.d.) source, Huffman code achieves the optimal average codeword length in the class of instantaneous code with a single code table. However, it is known that there exist time-variant encoders, which achieve a shorter average codeword length than the Huffman code, using multiple code tables and allowing at most k-bit decoding delay for k = 2, 3, 4, . . .. On the other hand, it is not known whether there exists a 1-bit delay decodable code, which achieves a shorter average length than the Huffman code. This paper proves that for a given i.i.d. source, a Huffman code achieves the optimal average codeword length in the class of 1-bit delay decodable codes with a finite number of code tables

Enzymatic control of anhydrobiosis-related accumulation of trehalose in the sleeping chironomid, Polypedilum vanderplanki

Larvae of an anhydrobiotic insect, Polypedilum vanderplanki, accumulate very large amounts of trehalose as a compatible solute on desiccation, but the molecular mechanisms underlying this accumulation are unclear. We therefore isolated the genes coding for trehalose metabolism enzymes, i.e. trehalose-6-phosphate synthase (TPS) and trehalose-6-phosphate phosphatase (TPP) for the synthesis step, and trehalase (TREH) for the degradation step. Although computational prediction indicated that the alternative splicing variants (PvTpsα/β) obtained encoded probable functional motifs consisting of a typical consensus domain of TPS and a conserved sequence of TPP, PvTpsα did not exert activity as TPP, but only as TPS. Instead, a distinct gene (PvTpp) obtained expressed TPP activity. Previous reports have suggested that insect TPS is, exceptionally, a bifunctional enzyme governing both TPS and TPP. In this article, we propose that TPS and TPP activities in insects can be attributed to discrete genes. The translated product of the TREH ortholog (PvTreh) certainly degraded trehalose to glucose. Trehalose was synthesized abundantly, consistent with increased activities of TPS and TPP and suppressed TREH activity. These results show that trehalose accumulation observed during anhydrobiosis induction in desiccating larvae can be attributed to the activation of the trehalose synthetic pathway and to the depression of trehalose hydrolysis