New nonexistence results on perfect permutation codes under the hamming metric

Xiang Wang; Wenjuan Yin

doi:10.3934/amc.2021058

Article Contents

2023, Volume 17, Issue 6: 1440-1452. Doi: 10.3934/amc.2021058

This issue Previous Article Rotational analysis of ChaCha permutation Next Article Generic constructions of MDS Euclidean self-dual codes via GRS codes

New nonexistence results on perfect permutation codes under the hamming metric

Xiang Wang^1, and
Wenjuan Yin^2, ,

1.
National Computer Network Emergency Response Technical Team/Coordination, Center of China (CNCERT/CC), Beijing, 100029, China
2.
School of Mathematical Sciences, Tiangong University, Tianjin, 300387, China

^* Corresponding author: Wenjuan Yin
^* Corresponding author: Wenjuan Yin

Received: March 2021

Revised: September 2021

Early access: December 2021

Published: December 2023

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Abstract

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 $ S_n $, the set of all permutations on $ n $ elements, under the Hamming metric. We prove the nonexistence of perfect $ t $-error-correcting codes in $ S_n $ under the Hamming metric, for more values of $ n $ and $ t $. Specifically, we propose some sufficient conditions of the nonexistence of perfect permutation codes. Further, we prove that there does not exist a perfect $ t $-error-correcting code in $ S_n $ under the Hamming metric for some $ n $ and $ t = 1,2,3,4 $, or $ 2t+1\leq n\leq \max\{4t^2e^{-2+1/t}-2,2t+1\} $ for $ t\geq 2 $, or $ \min\{\frac{e}{2}\sqrt{n+2},\lfloor\frac{n-1}{2}\rfloor\}\leq t\leq \lfloor\frac{n-1}{2}\rfloor $ for $ n\geq 7 $, where $ e $ is the Napier's constant.

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Mathematics Subject Classification: Primary: 94A15; Secondary: 68P30.

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Full Text(HTML)

Table 1. The values of $ (B^n(t),\frac{n!}{(n-t)!}) $ for $ 2\leq t\leq 4 $ and $ n\leq 10 $

$(B^n(t),\frac{n!}{(n-t)!})$	$n=5$	$n=6$	$n=7$	$n=8$	$n=9$	$n=10$
$t=2$	$(11,20)$	$(16,30)$	$(22,42)$	$(29,56)$	$(37,72)$	$(46,90)$
$t=3$	$-$	$-$	$(92,210)$	$(141,336)$	$(205,504)$	$(286,720)$
$t=4$	$-$	$-$	$-$	$-$	$(1339,3024)$	$(2176,5040)$

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Table 2. The values of $ n_0(t) $, $ Un_0(t) $, and $ Ln_0(t) $ for $ 2\leq t\leq 10 $, or $ t = 100,200 $

$t$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$100$	$200$
$n_0(t)$	$5$	$9$	$15$	$22$	$30$	$39$	$49$	$61$	$73$	$5659$	$22181$
$Un_0(t)$	$10.5$	$18.5$	$28.4$	$30.4$	$42.9$	$52.5$	$62.8$	$75.2$	$88.2$	$5819.4$	$22528.4$
$Ln_0(t)$	$5$	$7$	$9.1$	$14.5$	$21.0$	$28.6$	$36.9$	$46.9$	$57.9$	$5465.8$	$21760.2$

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