ON THE STRUCTURE OF HERMITIAN CODES

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 m(3) points of X-m rational over F-m2, each with multiplicity 1 . X-m has a cyclic group of automorphisms of order m(2) -1, which indu ces automorphisms of each of the the one-point algebraic geometric Gop pa 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 u sed to goad effect in both encoding and decoding. In this paper we exa mine the algebraic structure of these modules by means of the theory o f 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 f rom a. We also give a specialized algorithm for computing Groebner bas es, adapted to these particular modules. This algorithm has a much low er complexity than general Groebner basis algorithms, and has been suc cessfully implemented in the Maple computer algebra system. This permi ts the computation of Groebner bases and the construction of compact s ystematic encoders for some quite large codes (e.g. codes such as C-L( D,4010Q) on the curve X-16, With parameters n = 4096, k = 3891). (C) 1 997 Elsevier Science B.V.