On a Characterization of a State of Rank-Modulation Scheme Over Multi-Cell Ranking by a Group Action

Tomoharu SHIBUYA  Takeru SUDO  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E100-A   No.12   pp.2558-2571
Publication Date: 2017/12/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E100.A.2558
Type of Manuscript: Special Section PAPER (Special Section on Information Theory and Its Applications)
Category: Coding Theory and Techniques
rank-modulation,  coding for flash memories,  multi-cell ranking,  permutation matrices,  group action,  

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In this paper, we propose a group theoretic representation suitable for the rank-modulation (RM) scheme over the multi-cell ranking presented by En Gad et al. By introducing an action of the group of all permutation matrices on the set of all permutations, the scheme is clearly reformulated. Moreover, we introduce the concept of r-dominating sets over the multi-cell ranking, which is a generalization of conventional dominating sets, in the design of rank-modulation rewriting codes. The concept together with the proposed group theoretic representation yields an explicit formula of an upper bound on the size of the set of messages that can be stored in the memory by using RM rewriting codes over multi-cell ranking. This bound enables us to consider the trade-off between the size of the storable message set and the rewriting cost more closely. We also provide a concrete example of RM rewriting code that is not available by conventional approaches and whose size of the storable message set can not be achieved by conventional codes.