Constant-Weight Gray Codes for Local Rank Modulation

TL;DR

This paper explores constant-weight Gray codes within local rank-modulation schemes for flash memory, establishing conditions for their existence, analyzing weights 2 and 3, and constructing asymptotically-optimal codes for weight 3.

Contribution

It introduces necessary conditions for cyclic Gray codes in local rank-modulation and provides constructions for weight 3 codes that are asymptotically optimal.

Findings

No asymptotically-optimal cyclic codes of weight 2 exist.

Necessary conditions for cyclic Gray codes are established.

Constructed weight 3 codes are asymptotically optimal.

Abstract

We consider the local rank-modulation scheme in which a sliding window going over a sequence of real-valued variables induces a sequence of permutations. The local rank-modulation, as a generalization of the rank-modulation scheme, has been recently suggested as a way of storing information in flash memory. We study constant-weight Gray codes for the local rank-modulation scheme in order to simulate conventional multi-level flash cells while retaining the benefits of rank modulation. We provide necessary conditions for the existence of cyclic and cyclic optimal Gray codes. We then specifically study codes of weight 2 and upper bound their efficiency, thus proving that there are no such asymptotically-optimal cyclic codes. In contrast, we study codes of weight 3 and efficiently construct codes which are asymptotically-optimal.