Verão ICMC 2025

Projective Essential Idempotents: A connection between Algebra and Coding Theory

Modalidade: Formato presencial com transmissão online para possíveis participantes à distância.
Professor: Andre Luis dos Santos Duarte da Silva.
Duração: 10 horas (dias 27, 28, 29, 30 e 31 de janeiro de 2025).
Dias e horários: 27, 28, 29, 30 e 31 de janeiro de 2025, das 16:00 às 18:00 horas.
Local: As aulas ocorreram na sala 4-005 do ICMC.

Público-alvo: Graduandos e pós-graduandos em matemática ou áreas afins.

Programa:
In this mini-course, we introduce the concept of projective essential idempotents and explore their connection with coding theory. We discuss the existence problem and present some results about it. As a main application, we prove that every q-ary simplex code can be seen as an ideal of a twisted group algebra generated by a projective essential idempotent. We conclude by describing a decoding process using projective essential idempotents.

Bibliografia:
1. E. Berlekamp, Algebraic Coding Theory, World Scientific, 1968.
2. A. Duarte, Projective Essential idempotents, IEEE Transactions on Information Theory, v. 70, no. 8, pp. 5566-5572, 2024.
3. A. Duarte, R.A. Ferraz and C. Polcino Milies, Twisted group algebras of Abelian groups. Finite Fields and their Appl.. 95, 2024.
4. G. Karpilovsky, Group representations, Vol. 2, North-Holland, Amsterdam, 1993.
5. C. Polcino Milies, S.K. Sehgal, An introduction to Group Rings, Kluwer Academic Publishers, Dordrecht, 2002.
6. M. Shi, A. Alahmadi and P. Solé, Codes and Rings : Theory and Practice. Academic Press, 2017.