The generalized Goppa codes and related discrete designs from hermitian surfaces

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 (kin) and error correcting power (din) by invoking Riemann-Roch theorem and the subsequent discovery by Tsfasman, Vladut and Zink of a sequence of I inear codes in q symbols, whi ch performs better than those predicted by the Gi lbert--Varshamov bound for q ~ 49. Next, several new codes which have been constructed by embedding the non-degenerate Hermitian surface 3 3 3 3 0 of PG (3,4) , in a PG(9,4)Xo + xl + x2 + x3 = via monomials and weight-distributions of these codes are presented. 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, 63 8 73 63 ').. = 2-2 and v = 2, k = 2 + 2,).. = 2 + 2. In fact, the author has now shown that this construction can be extended to derive the Hadamard difference sets v = 22N+2, k = 22N+I _ 2N,).. = 22N _ 2N, v = 22N+2, k = 22N+I+ 2N,).. = 22N+ 2N. This will be reported in another paper. *Some of these results were presented at the 3~me Colloque International Thfiori