Bounds on the Probability of Undetected Error for <i>q</i>-Ary Codes

We study the probability of an undetected error for general <i>q</i>-ary codes. We give upper and lower bounds on this quantity, by the Linear Programming and the Polynomial methods, as a function of the length, size, and minimum distance. Sharper bounds are obtained in the important spe...

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Main Authors: Xuan Wang, Huizhou Liu, Patrick Solé
Format: Article
Language:English
Published: MDPI AG 2023-09-01
Series:Entropy
Subjects:
Online Access:https://www.mdpi.com/1099-4300/25/9/1349
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author Xuan Wang
Huizhou Liu
Patrick Solé
author_facet Xuan Wang
Huizhou Liu
Patrick Solé
author_sort Xuan Wang
collection DOAJ
description We study the probability of an undetected error for general <i>q</i>-ary codes. We give upper and lower bounds on this quantity, by the Linear Programming and the Polynomial methods, as a function of the length, size, and minimum distance. Sharper bounds are obtained in the important special case of binary Hamming codes. Finally, several examples are given to illustrate the results of this paper.
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spelling doaj.art-828b55c29e9b43948ca28c37f63e245b2023-11-19T10:36:22ZengMDPI AGEntropy1099-43002023-09-01259134910.3390/e25091349Bounds on the Probability of Undetected Error for <i>q</i>-Ary CodesXuan Wang0Huizhou Liu1Patrick Solé2School of Mathematical Sciences, Anhui University, Hefei 230601, ChinaState Grid Anhui Electric Power Co., Ltd., Hefei 230601, ChinaI2M, CNRS, Aix-Marseille Univetsity, Centrale Marseille, 13009 Marseilles, FranceWe study the probability of an undetected error for general <i>q</i>-ary codes. We give upper and lower bounds on this quantity, by the Linear Programming and the Polynomial methods, as a function of the length, size, and minimum distance. Sharper bounds are obtained in the important special case of binary Hamming codes. Finally, several examples are given to illustrate the results of this paper.https://www.mdpi.com/1099-4300/25/9/1349error correcting codesprobability of undetected errorlinear programmming
spellingShingle Xuan Wang
Huizhou Liu
Patrick Solé
Bounds on the Probability of Undetected Error for <i>q</i>-Ary Codes
Entropy
error correcting codes
probability of undetected error
linear programmming
title Bounds on the Probability of Undetected Error for <i>q</i>-Ary Codes
title_full Bounds on the Probability of Undetected Error for <i>q</i>-Ary Codes
title_fullStr Bounds on the Probability of Undetected Error for <i>q</i>-Ary Codes
title_full_unstemmed Bounds on the Probability of Undetected Error for <i>q</i>-Ary Codes
title_short Bounds on the Probability of Undetected Error for <i>q</i>-Ary Codes
title_sort bounds on the probability of undetected error for i q i ary codes
topic error correcting codes
probability of undetected error
linear programmming
url https://www.mdpi.com/1099-4300/25/9/1349
work_keys_str_mv AT xuanwang boundsontheprobabilityofundetectederrorforiqiarycodes
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