Coding Theory 1986: Coding Theory and Applications pp 116-124

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

• I. M. Chakravarti
Coding And Algebraic Geometry
Part of the Lecture Notes in Computer Science book series (LNCS, volume 311)

## 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 0 3 +x 1 3 +x 2 3 +x 3 3 =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=22N+2, k=22N+1−2N, λ=22N−2N, v=22N+2, k=22N+1+2N, λ=22N+2N. This will be reported in another paper.

## Keywords

Regular Graph Linear Code Association Scheme Parity Check Matrix Hermitian Variety
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. Ash, R.B. (1966) Information Theory, Interscience Publishers, John Wiley and Sons, New York.Google Scholar
2. Barlotti, A. (1965) Some Topics in Finite Geometrical Structures, Lecture Notes, Chapel Hill (Inst. of Statistics Mimeo Series no. 439).Google Scholar
3. Bose, R.C. (1961) On some connections between the design of experiments and information theory. Bull. Intern. Statist. Inst. 38, 257–271.Google Scholar
4. Bose, R.C. (1963) Some ternary error correcting codes and fractionally replicated designs. Colloques. Inter. CNRS, Paris, no. 110, 21–32.Google Scholar
5. Bose, R.C. and Mesner, D.M. (1959) On linear associative algebras corresponding to association schemes of partially balanced designs. Ann. Math. Statist. 30, 21–38.Google Scholar
6. Bose, R.C. and Chakravarti, I.M. (1966) Hermitian varieties in a finite projective space PG(N, q2), Canad. J. Math. 18, 1161–1182.Google Scholar
7. Bush, K.A. (1952) Orthogonal arrays of index unity. Ann. Math. Statist. 23, 426–434.Google Scholar
8. Chakravarti, I.M. (1963) Orthogonal and partially balanced arrays and their application in Design of Experiments, Metrika 7, 231–343.Google Scholar
9. Chakravarti, I.M. (1971) Some properties and applications of Hermitian varieties in PG(N,q2) in the construction of strongly regular graphs (two-class association schemes) and block designs. Journal of Comb. Theory, Series B, 11(3), 268–283.Google Scholar
10. Delsarte, P. (1972) Weights of linear codes and strongly regular normed spaces. Discrete Mathematics, 3, 47–64.Google Scholar
11. Delsarte, P. and Goethals, J.M. (1975) Alternating bilinear forms over GF(q). J. Combin. Theory. 19A Google Scholar
12. Dembowski, P. (1968) Finite Geometries, Springer-Verlag 1968.Google Scholar
13. Goppa, V.D. (1983) Algebraico-geometric codes. Math. USSR Izvestiya, 21(1), 75–91.Google Scholar
14. Goppa, V.D. (1984) Codes and information. Russian Math. Surveys., 39(1), 87–141.Google Scholar
15. Hill, R. (1978) Packing problems in Galois geometries over GF(3), Geometriae Dedicata, 7, 363–373.Google Scholar
16. MacWilliams, F.J. and Sloane, N.J.A. (1977) The Theory of Error-Correcting Codes, North Holland.Google Scholar
17. Robillard, P. (1969) Some results on the weight distribution of linear codes. IEEE Trans. Info. Theory 15, 706–709.Google Scholar
18. Raghavarao, D. (1971) Constructions and Combinatorial Problems in Design of Experiments. John Wiley and Sons, Inc., New York.Google Scholar
19. Segre (1965) Forme e geometrie hermitiane, con particolare riguardo al caso finito. Ann. Math. Pure Appl., 70 1, 202.Google Scholar
20. Segre, B. (1967) Introduction to Galois Geometries, Atti della Acc. Nazionale dei Lincei, Roma, 8(5), 137–236.Google Scholar
21. Singleton, R.C. (1964) Maximum distance q-nary codes. IEEE Trans. Info. Theory, 10, 116–118.Google Scholar
22. Tsfasman, M.A. (1982) Goppa codes that are better than the Varshamov-Gilbert bound. Problems of Info. Trans., 18, 163–165.Google Scholar
23. Tsfasman, M.A., Vladut, S.G. and Zink, T. (1982) Modular curves, Shimura curves and Goppa codes, better than Varshamov-Gilbert bound. Math. Nachr., 104, 13–28.Google Scholar
24. Vladut, S.G. and Drinfel'd, V.G. (1983) Number of points of an algebraic curve. Functional Anal, Appl. 17, 53–54.Google Scholar
25. Vladut, S.G., Katsman, G.L. and Tsfasman, M.A. (1984) Modular curve and codes with polynomial complexity of construction. Problems of Info. Transmission 20, 35–42.Google Scholar
26. Wolfmann, J. (1977) Codes projectifs a deux poids, "caps" complets et ensembles de differences. J. Combin. Theory, 23A, 208–222.Google Scholar