On perfect poset codes

Luciano Panek; Jerry Anderson Pinheiro; Marcelo Muniz Alves; Marcelo Firer

doi:10.3934/amc.2020061

Article Contents

2020, Volume 14, Issue 3: 477-489. Doi: 10.3934/amc.2020061

This issue Previous Article A shape-gain approach for vector quantization based on flat tori Next Article Challenge codes for physically unclonable functions with Gaussian delays: A maximum entropy problem

On perfect poset codes

1.
Center of Exact Sciences and Engineering, State University of West Paraná, Foz do Iguaçu, Paraná, Brazil
2.
ILACVN - Federal University for Latin American Integration, Foz do Iguaçu, Paraná, Brazil
3.
Polytechnic Center - Department of Mathematics, Federal University of Paraná, Curitiba, Paraná, Brazil
4.
IMECC - Department of Mathematics, University of Campinas, Campinas, São Paulo, Brazil

^* Corresponding author: Luciano Panek, luciano.panek@unioeste.br
^* Corresponding author: Luciano Panek, luciano.panek@unioeste.br

Received: November 30, 2018

Revised: August 31, 2019

Early access: January 2020

Published: August 2020

The second author was partially supported by São Paulo Research Foundation (FAPESP), grant 2017/10018-5. The third author was partially supported by CNPq, grant 306583/2016-0. The fourth author was partially supported by São Paulo Research Foundation (FAPESP), grant 2013/25977 and by CNPq, grant 304046/2017-5

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Abstract

We consider on $ \mathbb{F}_{q}^{n} $ metrics determined by posets and classify the parameters of $ 1 $-perfect poset codes in such metrics. We show that a code with same parameters of a $ 1 $-perfect poset code is not necessarily perfect, however, we give necessary and sufficient conditions for this to be true. Furthermore, we characterize the unique way up to a labeling on the poset, considering some conditions, to extend an $ r $-perfect poset code over $ \mathbb{F}_q^n $ to an $ r $-perfect poset code over $ \mathbb{F}_q^{n+m} $.

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Mathematics Subject Classification: Primary: 94B05, 06A06; Secondary: 20B30.

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Figure 1. Posets $ P $, $ Q $ and $ R $

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Figure 2. Posets $ P^+ $, $ Q^+ $ and $ R^+ $

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