Abstract
It is proved that a code L(q) which is monomially equivalent to the Pless symmetry code C(q) of length \(2q+2\) contains the (0,1)-incidence matrix of a Hadamard 3-\((2q+2,q+1,(q-1)/2)\) design D(q) associated with a Paley–Hadamard matrix of type II. Similarly, any ternary extended quadratic residue code contains the incidence matrix of a Hadamard 3-design associated with a Paley–Hadamard matrix of type I. If \(q=5, 11, 17, 23\), then the full permutation automorphism group of L(q) coincides with the full automorphism group of D(q), and a similar result holds for the ternary extended quadratic residue codes of lengths 24 and 48. All Hadamard matrices of order 36 formed by codewords of the Pless symmetry code C(17) are enumerated and classified up to equivalence. There are two equivalence classes of such matrices: the Paley–Hadamard matrix H of type I with a full automorphism group of order 19584, and a second regular Hadamard matrix \(H'\) such that the symmetric 2-(36, 15, 6) design D associated with \(H'\) has trivial full automorphism group, and the incidence matrix of D spans a ternary code equivalent to C(17).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
The symmetry code for \(q=5\) is equivalent to the extended ternary Golay code.
This is the order of the Paley–Hadamard matrix of Type II for \(q=17\) [7].
References
Assmus E.F. Jr., Key J.D.: Designs and Their Codes. Cambridge University Press, Cambridge (1992).
Assmus E.F. Jr., Key J.D.: Hadamard matrices and their designs: a coding-theoretic approach. Trans. Am. Math. Soc. 330(1), 269–293 (1992).
Assmus E.F. Jr., Mattson H.F. Jr.: New 5-designs. J. Combin. Theory Ser. A 6, 122–151 (1969).
Beth T., Jungnickel D., Lenz H.: Design Theory, 2nd edn Cambridge University Press, Cambridge (1999).
Bosma W., Cannon J.: Handbook of Magma Functions. Department of Mathematics, University of Sydney, Sydney (1994).
Bussemaker F.C., Tonchev V.D.: New extremal doubly-even codes of length 56 derived from Hadamard matrices of order 28. Discret. Math. 76, 45–49 (1989).
de Launey W., Stafford R.M.: On the automorphisms of Palye’s type II Hadamard matrix. Discret. Math. 308, 2910–2924 (2008).
Hall M. Jr.: Note on the Mathieu group \(M_{12}\). Arch. Math. 13, 334–340 (1962).
Hall M. Jr.: Combinatorial Theory. Wiley, New York (1986).
Harada M., Munemasa A.: A complete classification of ternary self-dual codes of length 24. J. Combin. Theory Ser. A 116, 1063–1072 (2009).
Huffman W.C.: On extremal self-dual ternary codes of lengths 28 to 40. IEEE Trans. Inf. Theory 38(4), 1395–1400 (1992).
Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).
Kantor W.M.: Automorphism groups of Hadamard matrices. J. Combin. Theory Ser. A 6, 279–281 (1969).
Lam C.W.H., Thiel L., Pautasso A.: On ternary codes generated by Hadamard matrices of order 24. Congr. Num. 89, 7–14 (1992).
Leon J.S., Pless V., Sloane N.J.A.: On ternary self-dual codes of length 24. IEEE Trans. Inf. Theory 27(2), 176–180 (1981).
Mallows C.L., Sloane N.J.A.: An upper bound for self-dual codes. Inf. Control 22, 188–200 (1973).
Nebe G.: On extremal self-dual ternary codes of length 48, Int. J. Combinat. https://doi.org/10.1155/2012/154281 (2012).
Nebe G., Villar D.: An analogue of the Pless symmetry codes. In: Seventh International Workshop on Optimal Codes and Related Topics, Albena, Bulgaria pp. 158–163 (2013).
Niskanen S., Östergård P.R.J.: Cliquer User’s Guide, Version 1.0. Tech. Rep. T48, Communications Laboratory, Helsinki University of Technology, Espoo, Finland (2003).
Norman C.W.: Nonisomorphic Hadamard designs. J. Combin. Theory Ser. A 21, 336–344 (1976).
Paley R.E.A.C.: On orthogonal matrices. J. Math. Phys. 12, 311–320 (1933).
Pless V.: On a new family of symmetry codes and related new five-designs. Bull. Am. Math. Soc. 75(6), 1339–1342 (1969).
Pless V.: Symmetry codes over \(GF(3)\) and new five-designs. J. Combin. Theory Ser. A 12, 119–142 (1972).
Pless V., Tonchev V.D.: Self-dual codes over \(GF(7)\). IEEE Trans. Info. Theory 33, 723–727 (1987).
Rains E.M., Sloane N.J.: Self-dual codes. In: Pless V.S., Huffman W.C. (eds.) Handbook of Coding Theory. Elsevier, Amsterdam (1998).
Tonchev V.D.: Hadamard matrices of order 28 with an automorphism of order 13. J. Combin. Theory Ser. A 35, 43–57 (1983).
Tonchev V.D.: Hadamard matrices of order 28 with an automorphism of order 7. J. Combin. Theory Ser. A 40, 62–81 (1985).
Tonchev V.D.: Hadamard matrices of order 36 with automorphisms of order 17. Nogoya Math. J. 104, 163–174 (1986).
Tonchev V.D.: Self-orthogonal designs and extremal dobly-even codes. J. Combin. Theory Ser. A 52, 197–205 (1989).
Tonchev V.D.: Combinatorial Configurations. Wiley, New York (1998).
Tonchev V.D.: Codes and designs. In: Pless V.S., Huffman W.C. (eds.) Handbook of Coding Theory, pp. 1229–1268. Elsevier, Amsterdam (1998).
Acknowledgements
The author thanks Cary Huffman for reading a preliminary version of this paper and making several useful suggestions that led to an improvement of the text.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue: On Coding Theory and Combinatorics: In Memory of Vera Pless”
Appendix
Appendix
A 2-(36, 15, 6) design associated with the Pless symmetry code of length 36
Rights and permissions
About this article
Cite this article
Tonchev, V.D. On Pless symmetry codes, ternary QR codes, and related Hadamard matrices and designs. Des. Codes Cryptogr. 90, 2753–2762 (2022). https://doi.org/10.1007/s10623-021-00941-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-021-00941-0