Finite geometries and linear codes

Finite geometry has relevant applications, especially in the theory of the linear correcting codes. A linear code of length n and dimension k over a finite field F_q is a k-dimensional subspace of the n-dimensional vector space over F_q . Here we concern geometric objects which are known to provide linear codes with good parameters. Finite planes are well known such objects. We focused on translation planes whose translation complement modulo scalars contains a subgroup isomorphic to A_6 . Another family of linear codes with good performance are the near-MDS codes. These codes arise from geometric objects generalizing arcs in higher dimensional projective spaces. We have provided a classification of near-MDS codes [12,6] over F_5 . Further geometric objects suitable for applications are the Hermitian varieties and their generalizations. We give a survey of our results on functional codes on Hermitian surfaces in PG(3,q^2).