Abstract
The 2-rank of any 2-(28,4,1) design (unital on 28 points) is known to be between 19 and 27. It is shown by the enumeration and analysis of certain binary linear codes that there are no unitals of 2-rank 20, and that there are exactly 4 isomorphism classes of unitals of 2-rank 21. Combined with previous results, this completes the classification of unitals on 28 points of 2-rank less than 22.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E. F. Assmus, Jr., On 2-ranks of Steiner triple systems, Electronic J. of Combinatorics, Vol. 2 (1995) Paper #R9.
E. F. Assmus, Jr., and J. D. Key, Designs and their Codes, Cambridge University Press, Cambridge (1992).
A. H. Baartmans, I. Landjev and V. D. Tonchev, On the binary codes of Steiner triple systems, Designs, Codes and Cryptography, Vol. 8 (1996) pp. 29-43.
Th. Beth, D. Jungnickel, and H. Lenz, Design Theory, Cambridge University Press (1986).
A. E. Brouwer, Some unitals on 28 points and their embedding in projective planes of order 9, in: Geometries and Groups, (M. Aigner and D. Jungnickel eds.), Lecture Notes in Mathematics, vol. 893 (1981) pp. 183-188.
A. E. Brouwer, The linear programming bound for binary linear codes, IEEE Trans. Information Theory, Vol. 39 (1993) pp. 677-680.
S. M. Dodunekov, S. B. Encheva, S. N. Kapralov, On the [28,7,12] binary self-complementary codes and their residuals, Designs, Codes and Cryptography, Vol. 4 (1994) pp. 57-67.
D. B. Jaffe, Binary linear codes: new results on nonexistence, preprint, accessible over the World Wide Web via http://www.math.unl.edu/~djaffe/#coding.
D. Jungnickel and V. D. Tonchev, On symmetric and quasi-symmetric designs with the symmetric difference property and their codes, J. Combin. Theory A, Vol. 59 (1992) pp. 40-50.
J. D. Key, B. Novick and F. E. Sullivan, Binary codes of structures dual to unitals, Congressus Numerantium (submitted).
H. Lüneburg, Some remarks concerning the Ree group of type (G2), J. Algebra, Vol. 3 (1966) pp. 256-259.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, New York (1977).
G. McGuire, V. D. Tonchev, and H. N. Ward, Proof of a conjecture of Brouwer concerning unitals on 28 points, Designs, Codes and Cryptography (to appear).
T. Penttila and G. Royle, Sets of type (m, n) in the affine and projective planes of order nine, Designs, Codes and Cryptography, Vol. 6 (1995) pp. 229-245.
V. Pless, Parents, Children, neighbors and the shadow, Contemporary Mathematics, Vol. 168 (1994) pp. 279-290.
V. D. Tonchev, Combinatorial Configurations, Longman, Wiley, New York (1988).
V. D. Tonchev, Unitals in the Hölz design on 28 points, Geometriae Dedicata, Vol. 38 (1991) pp. 357-363.
V. D. Tonchev, Quasi-symmetric designs, codes, quadrics, and hyperplane sections, Geometriae Dedicata, Vol. 48 (1993) pp. 295-308.
V. D. Tonchev, Codes, in: The CRC Handbook of Combinatorial Designs, (C. J. Colbourn and J. H. Dinitz eds.), CRC Press, New York (1996) pp. 517-543.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Jaffe, D.B., Tonchev, V.D. Computing Linear Codes and Unitals. Designs, Codes and Cryptography 14, 39–52 (1998). https://doi.org/10.1023/A:1008204420606
Issue Date:
DOI: https://doi.org/10.1023/A:1008204420606