Toric codes and finite geometries

Abstract

The theory of affine geometries over the rings Z/〈q−1〉 can be used to understand the properties of toric and generalized toric codes over Fq. The standard generator matrices of these codes are produced by evaluating collections of monomials in m variables at the points of the algebraic torus (Fq⁎)m. The exponent vector of such a monomial can be viewed as a point in one of these affine geometries and the minimum distance of the resulting code is strongly tied to the lines in the finite geometry that contain those points. We argue that this connection is, in fact, even more direct than the connection with the lattice geometry of those exponent vectors considered as elements of Z2 or R2. This point of view should be useful both as a way to visualize properties of these codes and as a guide to heuristic searches for good codes constructed in this fashion. In particular, we will use these ideas to see a reason why these constructions have been so successful over the field F8, but less successful in other cases.