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

Wang, Xiang; Fu, Fang-Wei

doi:10.1007/s10623-019-00693-y

Cited by 7 publications

(2 citation statements)

References 22 publications

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“…Construction A improves this size by a factor of 1 2 n 2 + 1 n 2 , times ρ ⌊n/2⌋+1 when n ≡ 2 (mod 4) (in the case of n ≡ 1 (mod 4) ρ ⌊n/2⌋+1 is eliminated by changing the order of congruence classes in σ 0 ). We note that a similar improvement was made concurrently by [35] in a preprint devoted solely to the case of d = 2, i.e., snake-in-the-box codes.…”

Section: Lemma 19mentioning

confidence: 65%

Limited-Magnitude Error-Correcting Gray Codes for Rank Modulation

Yehezkeally

Schwartz

2017

IEEE Trans. Inform. Theory

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We construct Gray codes over permutations for the rank-modulation scheme, which are also capable of correcting errors under the infinity-metric. These errors model limited-magnitude or spike errors, for which only single-error-detecting Gray codes are currently known. Surprisingly, the error-correcting codes we construct achieve a better asymptotic rate than that of presently known constructions not having the Gray property, and exceed the Gilbert-Varshamov bound. Additionally, we present efficient ranking and unranking procedures, as well as a decoding procedure that runs in linear time. Finally, we also apply our methods to solve an outstanding issue with error-detecting rank-modulation Gray codes (snake-in-the-box codes) under a different metric, the Kendall $\tau$-metric, in the group of permutations over an even number of elements $S_{2n}$, where we provide asymptotically optimal codes.Comment: Revised version for journal submission. Additional results include more tight auxiliary constructions, a decoding shcema, ranking/unranking procedures, and application to snake-in-the-box codes under the Kendall tau-metri

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Section: Lemma 19mentioning

confidence: 65%

Limited-Magnitude Error-Correcting Gray Codes for Rank Modulation

Yehezkeally

Schwartz

2017

IEEE Trans. Inform. Theory

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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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Nonexistence of perfect permutation codes under the $$\ell _{\infty }$$-metric

Wang

Yin

2022

AAECC

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

Cited by 7 publications

References 22 publications

Limited-Magnitude Error-Correcting Gray Codes for Rank Modulation

Limited-Magnitude Error-Correcting Gray Codes for Rank Modulation

New nonexistence results on perfect permutation codes under the hamming metric

Nonexistence of perfect permutation codes under the $$\ell _{\infty }$$-metric

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