Metrics in coding theory

• Grant number: 16/01551-9
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: June 01, 2016
• End date: March 31, 2019
• Field of knowledge: Engineering - Electrical Engineering - Telecommunications
• Principal Investigator: Marcelo Firer
• Grantee: Jerry Anderson Pinheiro
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Associated research grant: 13/25977-7 - Security and reliability of Information: theory and practice, AP.TEM
• Associated scholarship(s): 17/10018-5 - Alphabet size: vector network coding outperforms scalar network coding, BE.EP.PD

Abstract

Metrics are mathematical structures used in coding theory since the beginning of this theory. Its use is motivated by the possibility to obtain efficient decoding algorithms having same decision criteria as the ones determined by MAP decoders, which are the ideal decoders in terms of decision criteria. The interest in metrics based on partial orders in coding theory arises from the appearance of the Brualdi and Lawrence´s paper "Codes with a poset metric" in 1995. Such metrics are determined by weights, therefore are invariant by translations, a property necessary for syndrome decoding (in general, the most efficient known algorithm) to be a metric decoding criteria. Coding theory invariants, such as minimum distance, packing radius and covering radius, has been studied in vector spaces endowed with the so called poset metrics (metrics determined by partial orders). Generalizations of such metrics have been widely proposed. In this project, we propose to study the coding invariants for some of the poset metrics generalizations, as well as the possibility to characterize the decision criteria obtained from metrics preserving support. Poset metrics generalizations containing the family of combinatorial metrics, defined by Gabidulin in 1970, are the front-runner to the obtention of such characterization. In terms of applications, the poset metrics does not stand out, because channels matching these metrics (the ones in which metric and MAP decoders coincides) are purely theoretical, so they are not used in practice. The possibility to use such metrics in unequal error protection seems to be promising and will be exploited in this project. (AU)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

MACHADO, ROBERTO ASSIS; PINHEIRO, JERRY ANDERSON; FIRER, MARCELO. Characterization of Metrics Induced by Hierarchical Posets. IEEE TRANSACTIONS ON INFORMATION THEORY, v. 63, n. 6, p. 3630-3640, JUN 2017. (15/11286-8, 16/01551-9, 13/25977-7)