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

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.