A Family of Antipodal Distance-Regular Graphs Related to the Classical Preparata Codes
D. de Caen
, R. Mathon
and G.E. Moorhouse
DOI: 10.1023/A:1022429800058
Abstract
A new family of distance-regular graphs is constructed. They are antipodal 2 2 t-1-fold covers of the complete graph on 2 2 t vertices. The automorphism groups are determined, and the extended Preparata codes are reconstructed using walks on these graphs.
There are connections to other interesting structures: the graphs are equivalent to certain generalized Hadamard matrices; and their underlying 3-class association scheme is formally dual to the scheme of a system of linked symmetric designs obtained from Kerdock sets of skew matrices in characteristic two.
Pages: 317–327
Full Text: PDF
References
1. R.D. Baker, J.H. van Lint, and R.M. Wilson, "On the Preparata and Goethals codes," IEEE Trans. Info. Th. 29 (1983), 342-345.
2. B. Bollobas, Random Graphs, Academic Press, 1985.
3. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer-Veriag, 1989.
4. P.J. Cameron and J.H. van Lint, Designs, Graphs, Codes and their Links, Cambridge University Press, 1991.
5. A. Gardiner, "Antipodal Graphs of Diameter Three," Linear Algebra and its Applications 46 (1982), 215-219.
6. C.D. Godsil and A.D. Hensel, "Distance regular covers of the complete graph," Journal of Combin. Th. Ser. B 56 (1992), 205-238.
7. A.R. Mammons, Jr., P.V. Kumar, A.R. Calderbank, N.J.A. Sloane, and P. Sote, "The Z4-linearity of Kerdock, Preparata, Goethals and Related Codes," IEEE Trans. Inform. Theory 40 (1994), 301-319.
8. D. Jungnickel, "On difference matrices, resolvable transversal designs and generalized Hadamard matrices," Math. Z. 167 (1969), 49-60.
9. W.M. Kantor, "On the inequivalence of generalized Preparata codes," IEEE Trans. Info. Th. 29 (1983), 345- 348.
10. R. Mathon, "The systems of linked 2-(16,6,2) designs," Ars Combinatoria 11 (1981), 131-148.
2. B. Bollobas, Random Graphs, Academic Press, 1985.
3. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer-Veriag, 1989.
4. P.J. Cameron and J.H. van Lint, Designs, Graphs, Codes and their Links, Cambridge University Press, 1991.
5. A. Gardiner, "Antipodal Graphs of Diameter Three," Linear Algebra and its Applications 46 (1982), 215-219.
6. C.D. Godsil and A.D. Hensel, "Distance regular covers of the complete graph," Journal of Combin. Th. Ser. B 56 (1992), 205-238.
7. A.R. Mammons, Jr., P.V. Kumar, A.R. Calderbank, N.J.A. Sloane, and P. Sote, "The Z4-linearity of Kerdock, Preparata, Goethals and Related Codes," IEEE Trans. Inform. Theory 40 (1994), 301-319.
8. D. Jungnickel, "On difference matrices, resolvable transversal designs and generalized Hadamard matrices," Math. Z. 167 (1969), 49-60.
9. W.M. Kantor, "On the inequivalence of generalized Preparata codes," IEEE Trans. Info. Th. 29 (1983), 345- 348.
10. R. Mathon, "The systems of linked 2-(16,6,2) designs," Ars Combinatoria 11 (1981), 131-148.
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