Perfect codes in the l_p metric

• Grant number: 19/14390-1
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: September 01, 2019
• End date: February 28, 2021
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Grasiele Cristiane Jorge
• Grantee: Lucas Eduardo Nogueira Gonçalves
• Host 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.