The generalized Goppa codes and related discrete designs from hermitian surfaces in PG(3, s²)

Abstract

A short description is first given of the fascinating use of Hermitian curves and normal rational curves by Goppa in the construction of linear error correcting codes and estimation of their transmission rate (k/n) and error correcting power (d/n) by invoking Riemann-Roch theorem and the subsequent discovery by Tsfasman, Vladut and Zink of a sequence of linear codes in q symbols, which performs better than those predicted by the Gilbert-Varshamov bound for q ≥ 49.

Next, generalizing Goppa's construction (Goppa 1983, pp. 76–78), several new codes have been constructed by embedding the non-degenerate Hermitian surface x ³₀ +x ³₁ +x ³₂ +x ³₃ =0 of PG(3, 4), in a PG(9, 4) via monomials and the weight-distributions of these codes have been calculated.

Using the geometry of intersections of a non-degenerate Hermitian surface in PG(3, s²), by secant and tangent hyperplanes, a family of two-weight projective linear codes have been derived. For s=2, it is shown that the strongly regular graph of this code gives rise to the Hadamard difference sets v=2⁸, k=2⁷-2³, λ=2⁶-2³ and v=2⁸, k= 2⁷+2³, λ=2⁶+2³. In fact, the author has now shown that this construction can be extended to derive the Hadamard difference sets v=2^2N+2, k=2^2N+1−2^N, λ=2^2N−2^N, v=2^2N+2, k=2^2N+1+2^N, λ=2^2N+2^N. This will be reported in another paper.

Some of these results were presented at the 3ème Colloque International Théorie des Graphes et Combinatoire, Marseille-Luminy, 23–28 Juin 1986 and at the International Conference on Information Processing and Management of Uncertainty in Knowledge-based Systems, Paris, June 30–July 4, 1986.