Snake-in-the-Box Codes for Rank Modulation

TL;DR

This paper develops snake-in-the-box codes for rank modulation in flash memory, capable of error detection under two metrics, with constructions that achieve high rates and efficient encoding functions.

Contribution

It introduces new snake-in-the-box code constructions for rank modulation with asymptotically optimal rates and provides efficient algorithms for encoding and decoding.

Findings

Constructed snake-in-the-box codes with rate approaching 1

Provided efficient successor, ranking, and unranking functions

Analyzed bounds on code parameters

Abstract

Motivated by the rank-modulation scheme with applications to flash memory, we consider Gray codes capable of detecting a single error, also known as snake-in-the-box codes. We study two error metrics: Kendall's $\tau$-metric, which applies to charge-constrained errors, and the $\ell_\infty$-metric, which is useful in the case of limited magnitude errors. In both cases we construct snake-in-the-box codes with rate asymptotically tending to 1. We also provide efficient successor-calculation functions, as well as ranking and unranking functions. Finally, we also study bounds on the parameters of such codes.