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Perfect codes in the l_p metric

Grant number: 19/14390-1
Support type:Scholarships in Brazil - Scientific Initiation
Effective date (Start): September 01, 2019
Effective date (End): February 28, 2021
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Algebra
Principal researcher:Grasiele Cristiane Jorge
Grantee:Lucas Eduardo Nogueira Gonçalves
Home Institution: Instituto de Ciência e Tecnologia (ICT). Universidade Federal de São Paulo (UNIFESP). Campus São José dos Campos. São José dos Campos , SP, Brazil
Associated research grant:13/25977-7 - Security and reliability of Information: theory and practice, AP.TEM

Abstract

The Error Correcting Codes Theory was founded in the 1940s, considering codes in the Hamming metric. The Lee metric was first used in error correcting codes in 1958. In 1970 it was conjectured that there are no perfect codes in the Lee metric in dimensions greater or equal to 3 and with packing radius greater ou equal to 2. Although some results have already been demonstrated towards to the conjecture's proof, it is still far from being solved and has stimulated many works in the branch. For sufficiently large alphabets, linear perfect codes in Z_q^n in the Lee metric are relate to lattices in Z^n which are linear perfect codes in the metric sum (l_1 metric). Recently, linear perfect codes were considered in Z^n in the l_p metric, 1 \leq p <\infty. In this research project we propose the study of some papers that consider linear perfect codes in Z^n in the l_p metric to 1 \leq p <\infty.

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