Toric Codes over Finite Fields

In this note, a class of error-correcting codes is associated to a toric variety defined over a finite field [inline-graphic not available: see fulltext]q, analogous to the class of AG codes associated to a curve. For small q, many of these codes have parameters beating the Gilbert-Varshamov bound. In fact, using toric codes, we construct a (n,k,d)=(49,11,28) code over [inline-graphic not available: see fulltext]8, which is better than any other known code listed in Brouwer’s tables for that n, k and q. We give upper and lower bounds on the minimum distance. We conclude with a discussion of some decoding methods. Many examples are given throughout.