On the structure of Hermitian codes
Let \(X_m\) denote the Hermitian curve \(x^{m+1}= y^m +y\) over the field \(F_{m^2}\). The projective closure of the curve is smooth of degree \(m+1\) and genus \(m(m-1)/2\). The curve has \(m^3 +1\) rational points over \(F_{m^2}\), and achieves the Weil upper bound. The automorphism group of the curve is of order \(m^3 (m^3 +1)(m^2 -1)\) and contains a cyclic subgroup generated by \(\sigma: x \mapsto {\alpha}x, y \mapsto {\alpha}^{m+1} y\). Clearly \(\sigma\) fixes the point at infinity \(Q\) and permutes the other \(m^3\) rational points in \(m+2\) orbits. The Goppa code \(C_L (D,G)\) over the field \(F_q\) is defined as \[ C_L (D,G)= \{( f(P_1 ), \cdots , f(P_n )) \in F_q^n \mid f \in L(G) \} \] where \(D= \sum_{i=1}^n P_i\) and \(G\) are \(F_q\) rational divisors on a smooth curve \(X\). For \(X_m\) take \(D\) to be the sum of the \(m^3\) \(F_{m^2}\) rational points of \(X_m\) and \(G=aQ\) for some \(a \geq 1\) to obtain the Hermitian codes. The codes have length \(n=m^3\), dimension \(k= \text{dim} L(aQ) - \text{dim} L(aQ -D)\) and known distance, where \(L(G)\) is the space of rational functions defined over \(X_m\) whose divisor of poles is bounded by \(G\). It is shown in this work how the noted symmetries of the Hermitian curves can be exploited to achieve efficiencies in the encoding and decoding of the codes. It is first shown how to construct a root diagram for any code with an automorphism of order \(q-1\). This root diagram may be viewed as a generalization of the set of roots of the generator of a cyclic code of blocklength \(q-1\) over \(F_q\). This notion is then specialized to the case of Hermitian curves to give a root diagram of \(C_L (D,aG)\) in a purely combinatorial manner, for a given \(a\). Given the root diagram, it is shown how a Gröbner basis may be determined in a direct manner. From this approach, a compact systematic encoder can be constructed for very large codes. The relationship between the root diagram of a code \(C_L (D, aQ)\) and that of its dual is studied, and generalizations of known facts about cyclic codes of length \(q-1\) obtained. The paper concludes with a detailed consideration of the code \(C_L (D, 19Q)\) to illustrate the results.
scientific article; zbMATH DE number 1089295
scientific article