Algebraic structure of (generalized) toric codes

Building on the recent work surrounding toric codes, introduced in 2000 by J. Hansen, we further investigate the properties of this interesting class of error correcting cyclic codes. A toric code C is generated by creating monomials from a set of lattice points P in dimension m, and evaluating each of those monomials over all m-tuples of non-zero elements in a finite field of size q. Just as “ordinary” cyclic codes can be studied via properties of polynomials in one variable, we show that toric codes, which are m-dimensional cyclic codes, can be studied via m-variable polynomials. We aim in our work to generalize explicitly what the algebraic structure is for toric codes. In particular, we give formulas for finding the roots of (generalized) toric codes and their dual codes, and from these roots we derive a formula for an idempotent polynomial that generates the toric code.