A Systematic Census of Generalized Toric Codes over F4, F5 and F16

Toric codes are a specific class of linear codes, originally introduced by J. Hansen. In this report, we study generalized toric codes, which are generated by a set of points in Z m q−1. The orbits of these sets of points determine equivalent codes. In order to find and distinguish the codes for a given blocklength and dimension, we used various Magma processes to compute minimum distances and weight distributions of codes. There is an online table that contains much of the existing knowledge about the minimum distances of linear codes with certain dimensions at http://www.codetables.de. In our analysis of codes, we sought to find codes over F4 and F5 that would have minimum distances that exceed the lower bound listed in the online table, and thus would be the best known codes in existence for given parameters. In the process, we also noticed an interesting property about the average weights of words in toric codes and found codes from F5 and F16 that we believe to be interesting.