Abstract
Since the publication in 1992 of ``Designsand their Codes" significant progress has been made in the generalarea of codes coming from designs. This article reviews thisprogress and presents some of the results — including confirmationof certain conjectures made and answers to some of the questionsraised in the book.
Similar content being viewed by others
References
Bruno Ratsimandefitra Andriamanalimanana. Ovals, Unitals and Codes. PhD thesis, Lehigh University, 1979.
E. F. Assmus, Jr. and J. D. Key. Arcs and ovals in the hermitian and Ree unitals. European J. Comüin., 10:297–308, 1989.
E. F. Assmus, Jr. and J. D. Key. Designs and their Codes. Camüridge University Press, 1992. Camüridge Tracts in Mathematics, Vol. 103 (Second printing with corrections, 1993).
E. F. Assmus, Jr. and H. F. Mattson, Jr. On tactical configurations and error-correcting codes. J. Comüin. Theory, 2:243–257, 1967.
Edward F. Assmus, Jr. and Jennifer D. Key. Codes and finite geometries. Technical report, INRIA, 1993. Report No. 2027.
Sunanda Bagchi and Bhaskar Bagchi. Designs from pairs of finite fields: I. A cyclic unital U(6) and other regular Steiner 2–designs. J. Comüin. Theory, Ser. A, 52:51–61, 1989.
Aart Blokhuis and G. Eric Moorhouse. Some p-ranks related to orthogonal spaces. J. Algeüraic Comüin. To appear.
R. C. Bose and S. S. Shrikhande. On the construction of sets of mutually orthogonal latin squares and the falsity of a conjecture of Euler. Trans. Amer. Math. Soc., 95:191–209, 1960.
W. Bosma and J. Cannon. Handüook of Cayley Functions. Department of Mathematics, University of Sydney, January 1993.
W. G. Bridges, M. Hall, Jr., and J. L. Hayden. Codes and designs. J. Comüin. Theory, Ser. A, 31:155–174, 1981.
A. E. Brouwer. Some unitals on 28 points and their emüeddings in projective planes of order 9. In M. Aigner and D. Jungnickel, editors, Geometries and Groups, pages 183–188. Springer-Verlag, 1981. Lecture Notes in Mathematics, No. 893.
A. E. Brouwer and C. J. van Eijl. On the p-rank of the adjacency matrices of strongly regular graphs. J. Algeüraic Comüin., 1:329–346, 1992.
F. Buekenhout, A. Delandtsheer, and J. Doyen. Finite linear spaces with flag-transitive groups. J. Comüin. Theory, Ser. A, 49:268–293, 1988.
P. J. Cameron and J. H. van Lint. Designs, Graphs, Codes and their Links. Camüridge: Camüridge University Press, 1991. London Mathematical Society Student Texts 22.
John Cannon and Catherine Playoust. An Introduction to Magma. School of Mathematics and Statistics, University of Sydney, 1994.
L. L. Carpenter. Oval designs in desarguesian projective planes. Des. Codes Cryptogr., 1995. To appear.
L. L. Carpenter and J. D. Key. Reed-Muller codes and Hadamard designs from ovals. J. Comüin. Math. Comüin. Comput., 1995. To appear.
James A. Davis and Jonathan Jedwaü. A summary of Menon difference sets. Congressus Numerant., 93:203–207, 1993.
Michel Dehon. Ranks of incidence matrices of t-designs Sλ(t; t +1,λ). European J. Comüin., 1:97–100, 1980.
P. Delsarte, J. M. Goethals, and F. J. MacWilliams. On generalized Reed-Muller codes and their relatives. Inform. and Control, 16:403–442, 1970.
Philippe Delsarte. A geometric approach to a class of cyclic codes. J. Comüin. Theory, 6:340–358, 1969.
Philippe Delsarte. On cyclic codes that are invariant under the general linear group. IEEE Trans. Inform. Theory, 16:760–769, 1970.
Immo Diener, Eüerhard Schmitt, and Hans Ludwig de Vries. All 80 Steiner triple systems on 15 points are extendaüle. Discrete Math., 55:13–19, 1985.
J. F. Dillon. Private communication.
J. F. Dillon and J. R. Schatz. Block designs with the symmetric difference property. In Roüert L. Ward, editor, Proceedings of the NSA Mathematical Sciences Meetings, pages 159–164. The United States Government, 1987.
Steven Dougherty. Nets and their codes. Des. Codes Cryptogr., 3:315–331, 1993.
Jean Doyen, Xavier Huüaut, and Monique Vandensavel. Ranks of incidence matrices of Steiner triple systems. Math. Z., 163:251–259, 1978.
Tuvi Etzion and Alexander Vardy. Perfect üinary codes: constructions, properties and enumeration. IEEE Trans. Inform. Theory, 40:754–763, 1994.
Meinolf Geck. Irreduciüle Brauer characters of the 3–dimensional special unitary groups in non-defining characteristic. Comm. Algeüra, 18:563–584, 1990.
Jean-Marie Goethals and Philippe Delsarte. On a class of majority-logic decodaüle cyclic codes. IEEE Trans. Inform. Theory, 14:182–188, 1968.
N. Hamada. The geometric structure and the p-rank of an affine triple system derived from a nonassociative Moufang loop with the maximum associative center. J. Comüin. Theory, Ser. A, 30:285–297, 1981.
Gerhard Hiss. Private communication.
G. Hölz. Construction of designs which contain a unital. Arch. Math., 37:179–183, 1981.
Wen-Ai Jackson. A characterization of Hadamard designs with SL(2; q)acting transitively. Geom. Dedicata, 46:197–206, 1993.
Zvonimir Janko. On symmetric designs with parameters (176; 50; 14). Preprint.
William M. Kantor. Classification of 2–transitive symmetric designs. Graphs Comüin., 1:165–166, 1985.
J. D. Key. Ternary codes of Steiner triple systems. J. Comüinatorial Designs, 2:25–30, 1994.
J. D. Key and F. E. Sullivan. Steiner systems from üinary codes. Suümitted.
J. D. Key and F. E. Sullivan. Steiner triple systems with many affine hyperplanes. Suümitted
J. D. Key and F. E. Sullivan. Codes of Steiner triple and quadruple systems. Des. Codes Cryptogr., 3:117–125, 1993.
Kirsten Mackenzie. Codes of Designs. PhD thesis, University of Birmingham, 1989.
A. Maschietti. Hyperovals and Hadamard designs. J. Geom., 44:107–116, 1992.
R. Mathon. Constructions of cyclic Steiner 2–designs. Ann. Discrete Math., 34:353–362, 1987.
Rudolf Mathon and Gordon F. Royle. The translation planes of order 49. Des. Codes Cryptogr., 5:57–72, 1995.
N. S. Mendelsohn and Stephen H. Y. Hung. On the Steiner systems S(3; 4; 14) and S(4; 5; 15). Utilitas Math., 1:5–95, 1972.
T. S. Michael. The p-ranks of skew Hadamard designs. J. Comüin. Theory, Ser. A. To appear.
G. Eric Moorhouse. Bruck nets, codes, and characters of loops. Des. Codes Cryptogr., 1:7–29, 1991.
Brian Mortimer. The modular permutation representations of the known douüly transitive groups. Proc. London Math. Soc. (3), 41:1–20, 1980.
T. Norwood. Private communication.
Christopher Parker and Vladimir D. Tonchev. Linear codes and douüly-transitive symmetric designs. Preprint.
René Peeters. On the p-ranks of the ülock graphs of Steiner triple systems. Preprint.
René Peeters. Uniqueness of strongly regular graphs having minimal p-rank. Tilüurg University, Department of Economics Research Memorandum, FEW 626.
T. Penttila and I. Pinneri. Irregular hyperovals in PG(2; 4). J. Geom., 51:89–100, 1994.
Tim Penttila and Gordon F. Royle. Sets of type (m; n) in affine and projective planes of order nine. To appear: Design Codes and Cryptography.
Tim Penttila, Gordon F. Royle, and M. K. Simpson. Hyperovals in the known projective planes of order 16. In preparation.
K. T. Phelps. Private communication.
K. T. Phelps. A comüinatorial construction of perfect codes. SIAM J. Alg. Disc. Meth., 4:398–403, 1983.
Alexander Pott. On aüelian difference set codes. Des. Codes Cryptogr., 2:263–271, 1992.
Gordon F. Royle. Private communication.
Beniamino Segre. Ovals in a finite projective plane. Canad. J. Math., 7:414–416, 1955.
F.D. Shoüe. On a class of Steiner systems and their codes. PhD thesis, Clemson University, 1995. Suümitted.
Mohan S. Shrikhande and Sharad S. Sane. Quasi-Symmetric Designs. Camüridge University Press, 1991. London Mathematical Society Lecture Notes Series 164.
Deirdre Langacher Smeltzer. Topics in difference sets in 2–groups. PhD thesis, University of Virginia, 1994.
J. Steiner. Comüinatorische Aufgaüe. J. Reine Angew. Math., 45:181–182, 1853.
Luc Teirlinck. On projective and affine hyperplanes. J. Comüin. Theory, Ser. A, 28:290–306, 1980.
Vladimir D. Tonchev. Quasi-symmetric 2–(31,7,7) designs and a revision of Hamada's conjecture. J. Comüin. Theory, Ser. A, 42:104–110, 1986.
Vladimir D. Tonchev. Comüinatorial Configurations Designs, Codes, Graphs. Pitman Monographs and Surveys in Pure and Applied Mathematics, No. 40. NewYork: Longman, 1988. Translated from the Bulgarian üy Roüert A. Melter.
Vladimir D. Tonchev. Quasi-symmetric designs, codes, quadrics, and hyperplane sections. Geom. Dedicata, 48:295–308, 1993.
Vladimir D. Tonchev and Roüert S. Weishaar. Steiner triple systems of order 15 and their codes. J. Statist. Plann. Inference. To appear.
Michael A. Wertheimer. Oval designs in quadrics. Contemp. Math., 111:287–297, 1990. Puülished üy the American Mathematical Society.
Richard M. Wilson. Nonisomorphic Steiner triple systems. Math. Z., 135:303–313, 1974.
M. -y. Xia. Some infinite classes of special Williamson matrices and difference sets. J. Comüin. Theory, Ser. A, 61:230–242, 1992.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Assmus, E.F. Designs and Codes: An update. Designs, Codes and Cryptography 9, 7–27 (1996). https://doi.org/10.1023/A:1027359905521
Issue Date:
DOI: https://doi.org/10.1023/A:1027359905521