Shibuya Tomoharu

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, weintroduce the concept ofr-dominating sets over the multi-cell ranking, which is a generalization of conventional dominating sets, inthe 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.

In this paper, we study rank-modulation (RM) rewriting codes based on dominating sets. By introducing a special class of dominating sets in a construction of RM rewriting codes and employing the notion of a group action in an analysis of those codes, we succeeded to construct. RM rewriting codes based on the dominating sets that maximize the amount of stored data under the limit of rewriting cost for flash memories with 4 and 5 cells.

The amplitude damping (AD) quantum channel is one of the models describing evolution of quantum states. The construction of quantum error correcting codes for the AD channel based on classical codes has been presented, and Shor et al. proposed a class of classical codes over F-3 which are efficiently applicable to this construction. In this study, we expand Shor's construction to that over F-7, and succeeded to construct an AD code that has better parameters than AD codes constructed by Shor et al.