Group ring codes

• Grant number: 11/14340-2
• Support Opportunities: Scholarships in Brazil - Master
• Start date: March 01, 2012
• End date: January 31, 2013
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Edson Ryoji Okamoto Iwaki
• Grantee: Silvina Alejandra Alderete
• Host Institution: Centro de Matemática, Computação e Cognição (CMCC). Universidade Federal do ABC (UFABC). Ministério da Educação (Brasil). Santo André , SP, Brazil

Abstract

Error Correcting Codes Theory originated in 1948 with the work of Claude Shannon at the Bell's Labs. Boasting a wide range of applications in the present days, both scientific and technological. In essence, a Error Correcting Code is an organized way to add some data to each information that one wants to transmit or store, which allows us to retrieve information, detecting and correcting errors. Let G be a finite group, R be an associative ring with unity. Denote by RG the set of all finite formal sums of elements r_gg, with r_g \in R, g \in G, represented by \sum_{g \in G}r_gg. In case when the ring R is commutative, RG is called the group algebra of G over R.Let Fq denote a finite field, G be a finite group. A group ring code is an ideal of the group algebra FqG. In case the characteristic of the field Fq doesn´t divides the order of the group G, the group algebra FqG is semisimple and any group ring code is generated by an idempotent element of FqG. Moreover, any group ring code is a direct sum of minimal ideals of FqG. Several classes of codes can be considered as ideals of group algebras and in several classes of rings. So one becomes natural to study the following questions:*Determine the group ring codes satisfying some interesting property, for example, determine the group ring codes of the group algebra FqG which are minimal.*Determine the weight and the dimensions of the group ring codes of the group algebra FqG. (AU)