On the structure of Hermitian codes

Abstract

Let X m denote the Hermitian curve x m + 1 = y m + y over the field F m 2 . Let Q be the single point at infinity, and let D be the sum of the other m 3 points of X m rational over F m 2 , each with multiplicity 1. X m has a cyclic group of automorphisms of order m 2 − 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, 4010 Q) on the curve X 16, with parameters n = 4096, k = 3891).