Snake-in-the-box codes under the $\ell _{\infty }$-metric for rank modulation

Abstract

In the rank modulation scheme, Gray codes are very useful in the realization of flash memories. For a Gray code in this scheme, two adjacent codewords are obtained by using some “push-to-the-top” operations. Moreover, snake-in-the-box codes under the $\ell _{\infty }$-metric ($\ell _{\infty }$-snakes) are Gray codes, which can be capable of detecting one $\ell _{\infty }$-error. In this paper, we give two constructions of $\ell _{\infty }$-snakes. On the one hand, inspired by Yehezkeally and Schwartz’s construction, we present a new construction of the $\ell _{\infty }$-snake. The length of this $\ell _{\infty }$-snake is longer than the length of the $\ell _{\infty }$-snake constructed by Yehezkeally and Schwartz. On the other hand, we also give another construction of $\ell _{\infty }$-snakes by using ${\mathcal {K}}$-snakes and obtain the longer $\ell _{\infty }$-snakes than the previously known ones.