Constructions of Snake-in-the-Box Codes for Rank Modulation.

Snake-in-the-box code is a Gray code, which is capable of detecting a single error. Gray codes are important in the context of the rank modulation scheme, which was suggested recently for representing information in flash memories. For a Gray code in this scheme, the codewords are permutations, two consecutive codewords are obtained using the push-to-the-top operation, and distance measure is defined on permutations. In this paper, the Kendall’s \(\tau \) -metric is used as the distance measure. We present a general method for constructing such Gray codes. We apply the method recursively to obtain a snake of length \(M_{2n+1}=((2n+1)(2n)-1)M_{2n-1}\) for permutations of \(S_{2n+1}\) , from a snake of length \(M_{2n-1}\) for permutations of \(S_{2n-1}\) . Thus, we have \(\lim \limits _{n\to \infty }{M_{2n+1}}/{S_{2n+1}}\approx 0.4338\) , improving on the previous known ratio of \(\lim \limits _{n\to \infty }{1}/\sqrt {{({\pi n})}}\) . Using the general method, we also present a direct construction. This direct construction is based on necklaces and it might yield snakes of length \({(2n+1)!}/{2} -2n\,+\,1\) for permutations of \(S_{2n+1}\) . The direct construction was applied successfully for \(S_{7}\) and \(S_{9}\) , and hence \(\lim \limits _{n\to \infty } {M_{2n+1}}/{S_{2n+1}}\approx 0.4743\) . [ABSTRACT FROM PUBLISHER]