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

Abstract

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 T-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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2n+1</sub> = ((2n + 1)(2n) - 1)M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2n-1</sub> for permutations of S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2n+1</sub> , from a snake of length M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2n-1</sub> for permutations of S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2n-1</sub> . Thus, we have lim;n→∞ M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2n+1</sub> /S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2n+1</sub> ≈0.4338, improving on the previous known ratio of lim;n→∞ 1/√(πn). 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2n+1</sub> . The direct construction was applied successfully for S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">7</sub> and S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">9</sub> , and hence lim;n→∞ M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2n+1</sub> /S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2n+1</sub> ≈0.4743.