Conference abstracts

Session S07 - Finite Fields

No date set

## Projective Nested Cartesian Codes

### Universidade Federal de Uberlândia, Brasil   -   victor.neumann@ufu.br

In this work we introduce a new type of code, called projective nested cartesian code. It is obtained by the evaluation of homogeneous polynomials of a fixed degree on the set $$\left[A_0\times A_1\times\cdots\times A_n\right]:=\left\{(a_0:\cdots :a_n) \vert\, a_i\in A_i \mbox{ for all } i\right\}\subset\mathbb{P}^n(\mathbb{F}_q),$$ where $A_0,A_1,\ldots,A_n$ is a collection of non-empty subsets of $\mathbb{F}_q$ such that for all $i=0,\ldots,n$ we have $0\in A_i,$ and for every $i=1,\ldots,n$ we have $A_j A_{i-1} \subset A_j$ for $j=i,\ldots, n$. These codes may be seen as a generalization of the so-called projective Reed-Muller codes. We calculate the length and the dimension of such codes, a lower bound for the minimum distance and the exact minimum distance in the special case where the sets $A_i$ are subfields of $\mathbb{F}_q$ (so it includes the projective Reed-Muller codes).

Joint work with Cícero Carvalho (Universidade Federal de Uberlândia, Brasil) and Hiram López (Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, México).