Toric codes over finite fields

In this note, a class of error-correcting codes is associated to a toric variety associated to a fan defined over a finite field ${\text{fff}}_q$, analogous to the class of Goppa codes associated to a curve. For such a ``toric code’’ satisfying certain additional conditions, we present an efficient decoding algorithm for the dual of a Goppa code. Many examples are given. 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 ${\text{fff}}_8$, which is better than any other known code listed in Brouwer’s on-line tables for that $n$ and $k$.