The geometry of two-weight codes

Bulletin of the London Mathematical Society

Volumn 18, Issue 2, 1986, Pages 97-122

  Calderbank, R ^a   Kantor, W M ^b  

^a LUCENT TECHNOLOGIES   (United States)

^b UNIVERSITY OF OREGON   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001850013     PISSN: 00246093     EISSN: 14692120     Source Type: Journal    
DOI: 10.1112/blms/18.2.97     Document Type: Article

References (58)

1. M. R. BERLEKAMP, J. H. VAN LINT and J. J. SEIDEL, ‘A strongly regular graph derived from the perfect ternary Golay code’, A survey of combinatorial theory (ed. J. N. Srivastava, North-Hoiland, Amsterdam, 1973), pp. 25-30.
2. N. Buekers, ‘On the generalized Ramanujan-Nagell equation I’ Acta Arith. 38 (1981) 389-410.
3. R. C. Bose and D. M. Mesner, ‘On linear associative algebras corresponding to association schemes of partially balanced designs’, Ann. Math. Statist. 30 (1959) 21-38.
4. R. C. Bose and T. Shimamoto, ‘Classification and analysis of partially balanced incomplete block designs with two associate classes’, J. Amer. Statist. Assoc. 47 (1952) 151-184.
5. A. BREMNER, R. CALDERBANK, P. HANLON, P. MORTON and J. WOLFSKILL, ‘Two-weight ternary codes and the equation y2 = 4.3°+ 13’, J. Number Theory 16 (1983) 212-234.
6. A. A. BRUEN and J. W. P. HIRSCHFELD, ‘Applications of line geometry over finite fields II. The hermitian surface’, Geom. Dedicata 1 (1978) 333-353.
7. A. A. BRUEN and J. A. THAS, ‘Partial spreads, packings and hermitian manifolds in PG(3, g)’, Math. Z. 125 (1972) 122-128.
8. R. CALDERBANK, ‘On uniformly packed [n, n-k, 4] codes over GF(q) and a class of caps in PG(3, q)’ J. London Math. Soc. 26 (1982) 365-384.
9. P. J. CAMERON and J. H. VANLint, ‘On the partial geometry pg (6, 6, 2)’, J. Comb. Theory (14) 32 (1982) 252-255.
10. P. J. CAMERON, ‘Partial quadrangles’, Quart. J. Math. 26 (1975) 61-73.
11. P. CAMION, Difference sets in elementary abelian groups (Les Presses de L’Université de Montréal, 1979).
12. J. H. CONWAY, ‘Three lectures on exceptional groups’, Finite simple groups (ed. M. B. Powell and G. Higman, Academic Press, New York, 1971), pp. 215-247.
13. H. S. M. COXETER, ‘Twelve points in PG(5, 3) with 95040 self-transformations’, Proc. Royal Soc. (14) 247 (1958) 279-293.
14. P. DELSARTE, ‘Two-weight linear codes and strongly regular graphs’, Report R160, MBLE Res. Lab., Brussels, 1971.
15. P. DELSARTE, ‘Weights of linear codes and strongly regular normed spaces’, Discrete Math. 3 (1972) 47-64.
16. P. DELSARTE, ‘An algebraic approach to the association schemes of coding theory’, Philips Research Reports Supplements 10, 1973.
17. P. DELSARTE and J. M. GOETHALS, ‘Irreducible binary cyclic codes of even dimension’, Combinatorial mathematics and its applications, Proc. Second Chapel Hill Conference (Univ. of N. Carolina, Chapel Hill, N.C., 1970), pp. 100-113.
18. P. DELSARTE and J. M. GOETHALS, ‘Alternating bilinear forms over GF(q) J. Comb. Theory (A) 19 (1975) 26-50.
19. P. DEMBOWSKI, Finite geometries (Springer-Verlag, Berlin, 1968).
20. R. H. F. DENNISTON, ‘Some maximal arcs in finite projective planes’, J. Comb. Theory 6 (1969) 317-319.
21. L. E. DICKSON, Linear groups with an exposition of the Galois field theory (Dover, New York, 1958).
22. J. DIEUDONNE, La géométrie des groupes classiques (3-ème édition, Springer, Berlin, 1971).
23. J. F. DILLON, ‘Elementary Hadamard difference sets’, Ph.D. Thesis, Univ. of Maryland, 1974.
24. J. F. DILLON, ‘Elementary Hadamard difference sets’, Proc. 6 th S-E Confi Combinatorics, Graph Theory and Computing (Utilitas Math., Winnepeg, 1975), pp. 237-249.
25. S. M. DODUNEKOV and V. I. IORGOV, ‘Distribution of weights of irreducible cyclic codes and strongly regular graphs’, Problems Inform. Transmission 16 (1980) no. 2, 105-110.
26. M. DE FINIS, ‘On k-sets of type (m, n) in projective planes of square order’, Finite geometries and designs, L.M.S. Lecture Note Series 49 (Cambridge Univ. Press, 1981), pp. 98-103.
27. D. A. FOULSER, ‘Solvable primitive permutation groups of low rank’, Trans. Amer. Math. Soc. 143 (1969) 1-54.
28. R. A. GAMES, ‘The packing problem for finite projective geometries’, Thesis, The Ohio State University, 1980.
29. J. M. GOETHALS and H. C. A. VAN TILBORG, ‘Uniformly packed codes’, Philips Research Reports 30 (1975) 9-36.
30. R. Hill, ‘On the largest size cap in S5 3’ Rend. Accad. Naz. Lincei (8) 54 (1973) 378-384.
31. R. HILL, ‘Caps and groups’, Atti dei Convegni Lincei, Colloquio Internazionale sulle Teorie Combina-torie(Roma 1973), no. 17 (Accad. Naz. Lincei, 1976), pp. 384-394.
32. R. Hill, ‘Caps and codes’, Discrete Math. 22 (1978) 111-137.
33. J. W. P. HIRSCHFELD, Projective geometries over finite fields (Oxford Univ. Press, Oxford, 1979).
34. X. L. HUBAUT, ‘Strongly regular graphs’, Discrete Math. 13 (1975) 357-381.
35. W. M. KANTOR, ‘Symplectic groups, symmetric designs, and line ovals’, J. Algebra 33 (1975) 43-58.
36. W. M. KANTOR, ‘Spreads, translation planes, and Kerdock sets I’, SIAM J. Algebraic Discrete Methods 3 (1982) 151-165.
37. J. H. VAN LINT, ‘A survey of perfect codes’, Rocky Mountain J. Math. 5 (1975) 199-224.
38. J. H. VAN LINT and A. SCHRIJVER, ‘Construction of strongly regular graphs, two-weight codes and partial geometries by finite fields’, Combinatorica 1 (1981) 63-75.
39. H. LÜNEBURG, Translation planes (Springer, New York, 1980).
40. J. MCLAUGHLIN, ‘A simple group of order 898, 128, 000’, Theory of finite groups (Benjamin, New York, 1969), pp. 109-111.
41. F. J. MACWILLIAMS, ‘Combinatorial problems of elementary group theory’, Thesis, Department of Math., Harvard University, 1962.
42. F. J. MACWILLIAMS, ‘A theorem on the distribution of weights in a systematic code’, Bell System Tech. J. 42 (1963) 79-94.
43. F. J. MACWILLIAMS and N. J. A. SLOANE, The theory of error-correcting codes (North-Holland, Amsterdam, 1977).
44. H. B. MANN, ‘Difference sets in elementary abelian groups’, Illinois J. Math. 9 (1965) 212-219.
45. R. J. MCELIECE, ‘A class of two-weight codes’, J. P. L. Space Programs Summary, 37-41, Vol. IV, pp. 264-266.
46. R. MEZ, ‘On point sets with two intersection numbers’, unpublished report.
47. O. S. ROTHAUS, ‘On “bent” functions’, J. Comb. Theory (A) 20 (1976) 300-305.
48. B. SEGRE, ‘Forme e geometrie hermitiane, con particolare riguardo al caso finito’, Ann. Mat. Pura Appl. 70(1965) 1-202.
49. N. V. SEMAKOV, V. A. ZINOVJEV and G. V. ZAITZEV, ‘Uniformly packed codes’, Problems Inform. Transmission 7 (1971) no. 1, 30-39.
50. M. TALLINISCAFATI, ‘(k, n)-archi di un piano grafico finito, con particolare riguardo a quelli con due caratteri. Note I, II’, Atti Accad. Naz. Lincei Mem. CL Sei. Fis. Mat. Nat. (8) 40 (1966) 812-818, 1020-1025.
51. J. A. THAS, ‘A combinatorial problem’, Geom. Dedicata 1 (1973) 236-240.
52. A. TIETÄVÄINEN, ‘A short proof for the non-existence of unknown perfect codes over GF(-), q > 2’, Ann. Acad. Sei. Fenn. Ser. A I. Math. 580 (1974) 1-6.
53. H. C. A. VAN TILBORG, ‘Uniformly packed codes’, Thesis, Tech. Univ. Eindhoven, 1976.
54. J. Tits, ‘Ovoïdes et groupes de Suzuki’, Arch. Math. 13 (1962) 187-198.
55. J. WOLFMANN, ‘Formes quadratiques et codes à deux poids’, C. R. Acad. Sei. Paris 281 (1975) A533-A535.
56. J. WOLFMANN, ‘Codes projectifs à deux ou trois poids associés aux hyperquadriques d’une géométrie finie’, Discrete Math. 14 (1975) 185-211.
57. J. WOLFMANN, ‘Propriétés combinatoires déduites de l’étude des codes à deux poids’, C. R. Acad. Sei. Paris 284 (1977) A641-A642.
58. J. WOLFMANN, ‘Codes projectifs à deux poids, “caps” complets et ensembles de différences’, J. Comb. Theory (A) 23 (1977) 208-222.