On the snake-in-the-box codes for rank modulation under Kendall’s $$\tau $$ τ -metric

Wang, Xiang; Fu, Fang-Wei

doi:10.1007/s10623-016-0239-y

Cited by 10 publications

(4 citation statements)

References 16 publications

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“…Firstly, we will start this example with one initial permutation, denoted by σ0 . By (35), we have that σ0 = [2,1,3,5,7,4,6].…”

Section: B One Example Of ℓ ∞ -Snakes By Using K-snakesmentioning

confidence: 99%

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

Wang

2019

Des. Codes Cryptogr.

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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 one "push-to-the-top" operation. Moreover, snake-in-the-box codes under the ℓ∞-metric are Gray codes, which can be capable of detecting one ℓ∞-error. In this paper, we give two constructions of ℓ∞snakes. On the one hand, inspired by Yehezkeally and Schwartz's construction, we present a new construction of the ℓ∞-snake. The length of this ℓ∞-snake is longer than the length of the ℓ∞-snake constructed by Yehezkeally and Schwartz. On the other hand, we also give another construction of ℓ∞-snakes by using K-snakes and obtain the longer ℓ∞-snakes than the previously known ones.

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“…Firstly, we will start this example with one initial permutation, denoted by σ0 . By (35), we have that σ0 = [2,1,3,5,7,4,6].…”

Section: B One Example Of ℓ ∞ -Snakes By Using K-snakesmentioning

confidence: 99%

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

Wang

2019

Des. Codes Cryptogr.

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“…, n}. Permutation codes under various metrics have been studied, such as the ∞ -metric [18,26,29], the Ulam metric [12], the Kendall τmetric [1,17,25,30], and the Hamming metric [24,10,6,21,5,28]. Moreover, a survey on metrics related to permutations is given in [8].…”

Section: Introductionmentioning

confidence: 99%

New nonexistence results on perfect permutation codes under the hamming metric

Wang¹,

Yin²

2023

AMC

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<p style='text-indent:20px;'>Permutation codes under the Hamming metric are interesting topics due to their applications in power line communications and block ciphers. In this paper, we study perfect permutation codes in <inline-formula><tex-math id="M1">\begin{document}$ S_n $\end{document}</tex-math></inline-formula>, the set of all permutations on <inline-formula><tex-math id="M2">\begin{document}$ n $\end{document}</tex-math></inline-formula> elements, under the Hamming metric. We prove the nonexistence of perfect <inline-formula><tex-math id="M3">\begin{document}$ t $\end{document}</tex-math></inline-formula>-error-correcting codes in <inline-formula><tex-math id="M4">\begin{document}$ S_n $\end{document}</tex-math></inline-formula> under the Hamming metric, for more values of <inline-formula><tex-math id="M5">\begin{document}$ n $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ t $\end{document}</tex-math></inline-formula>. Specifically, we propose some sufficient conditions of the nonexistence of perfect permutation codes. Further, we prove that there does not exist a perfect <inline-formula><tex-math id="M7">\begin{document}$ t $\end{document}</tex-math></inline-formula>-error-correcting code in <inline-formula><tex-math id="M8">\begin{document}$ S_n $\end{document}</tex-math></inline-formula> under the Hamming metric for some <inline-formula><tex-math id="M9">\begin{document}$ n $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ t = 1,2,3,4 $\end{document}</tex-math></inline-formula>, or <inline-formula><tex-math id="M11">\begin{document}$ 2t+1\leq n\leq \max\{4t^2e^{-2+1/t}-2,2t+1\} $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M12">\begin{document}$ t\geq 2 $\end{document}</tex-math></inline-formula>, or <inline-formula><tex-math id="M13">\begin{document}$ \min\{\frac{e}{2}\sqrt{n+2},\lfloor\frac{n-1}{2}\rfloor\}\leq t\leq \lfloor\frac{n-1}{2}\rfloor $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M14">\begin{document}$ n\geq 7 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M15">\begin{document}$ e $\end{document}</tex-math></inline-formula> is the Napier's constant.</p>

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“…However, the application of permutation codes and multipermutation codes for use in non-volatile memory storage systems such as flash memory has received attention in the coding theory literature in recent years [1], [12], [13], [17], [21], [22]. One of the main distance metrics in the literature has been the Kendall-τ metric, which is suitable for correction of the type of error expected to occur in flash memory devices [5], [11], [12], [13], [23]. Errors occur in these devices when the electric cell charges storing information leak over time or there is an overshoot of charge level in the rewriting process.…”

Section: Introductionmentioning

confidence: 99%

Ulam Ball Size Analysis for Permutation and Multipermutation Codes Correcting Translocation Errors

Kong

2019

IEEE Trans. Inform. Theory

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Permutation and multipermutation codes in the Ulam metric have been suggested for use in non-volatile memory storage systems such as flash memory devices. In this paper we introduce a new method to calculate permutation sphere sizes in the Ulam metric using Young Tableaux and prove the non-existence of non-trivial perfect permutation codes in the Ulam metric. We then extend the study to multipermutations, providing tight upper and lower bounds on multipermutation Ulam sphere sizes and resulting upper and lower bounds on the maximal size of multipermutation codes in the Ulam metric.J. Kong is with the

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On the snake-in-the-box codes for rank modulation under Kendall’s $$\tau $$ τ -metric

Cited by 10 publications

References 16 publications

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

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

New nonexistence results on perfect permutation codes under the hamming metric

Ulam Ball Size Analysis for Permutation and Multipermutation Codes Correcting Translocation Errors

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