Let X_m denote the Hermitian curve x^{m + 1} = y^m + y over the field F_m². Let Q be the single point at infinity, and let D be the sum of the other m³ points of X_m rational over F_m², each with multiplicity 1. X_m has a cyclic group of automorphisms of order m² − 1, which induces automorphisms of each of the the one-point algebraic geometric Goppa codes C_L(D, aQ) and their duals. As a result, these codes have the structure of modules over the ring F_q[t], and this structure can be used to good effect in both encoding and decoding. In this paper we examine the algebraic structure of these modules by means of the theory of Groebner bases. We introduce a root diagram for each of these codes (analogous to the set of roots for a cyclic code of length q − 1 over F_q), and show how the root diagram may be determined combinatorially from a. We also give a specialized algorithm for computing Groebner bases, adapted to these particular modules. This algorithm has a much lower complexity than general Groebner basis algorithms, and has been successfully implemented in the Maple computer algebra system. This permits the computation of Groebner bases and the construction of compact systematic encoders for some quite large codes (e.g. codes such as C_L(D, 4010Q) on the curve X₁₆, with parameters n = 4096, k = 3891).