The moment map of a Lie group representation
HTML articles powered by AMS MathViewer
- by N. J. Wildberger
- Trans. Amer. Math. Soc. 330 (1992), 257-268
- DOI: https://doi.org/10.1090/S0002-9947-1992-1040046-6
- PDF | Request permission
Abstract:
Given an $m \times m$ Hadamard matrix one can extract ${m^2}$ symmetric designs on $m - 1$ points each of which extends uniquely to a $3$-design. Further, when $m$ is a square, certain Hadamard matrices yield symmetric designs on $m$ points. We study these, and other classes of designs associated with Hadamard matrices, using the tools of algebraic coding theory and the customary association of linear codes with designs. This leads naturally to the notion, defined for any prime $p$, of $p$-equivalence for Hadamard matrices for which the standard equivalence of Hadamard matrices is, in general, a refinement: for example, the sixty $24 \times 24$ matrices fall into only six $2$-equivalence classes. In the $16 \times 16$ case, $2$-equivalence is identical to the standard equivalence, but our results illuminate this case also, explaining why only the Sylvester matrix can be obtained from a difference set in an elementary abelian $2$-group, why two of the matrices cannot be obtained from a symmetric design on $16$ points, and how the various designs may be viewed through the lens of the four-dimensional affine space over the two-element field.References
- E. F. Assmus Jr., On the theory of designs, Surveys in combinatorics, 1989 (Norwich, 1989) London Math. Soc. Lecture Note Ser., vol. 141, Cambridge Univ. Press, Cambridge, 1989, pp.Β 1β21. MR 1036749
- E. F. Assmus Jr. and J. D. Key, Affine and projective planes, Discrete Math. 83 (1990), no.Β 2-3, 161β187. MR 1065696, DOI 10.1016/0012-365X(90)90003-Z
- E. F. Assmus Jr. and J. D. Key, Translation planes and derivation sets, J. Geom. 37 (1990), no.Β 1-2, 3β16. MR 1041974, DOI 10.1007/BF01230354
- Chester J. Salwach and Joseph A. Mezzaroba, The four known biplanes with $k=11$, Internat. J. Math. Math. Sci. 2 (1979), no.Β 2, 251β260. MR 539202, DOI 10.1155/S0161171279000235
- Bhaskar Bagchi and N. S. Narasimha Sastry, Even order inversive planes, generalized quadrangles and codes, Geom. Dedicata 22 (1987), no.Β 2, 137β147. MR 877206, DOI 10.1007/BF00181262
- Vasanti N. Bhat and S. S. Shrikhande, Non-isomorphic solutions of some balanced incomplete block designs. I, J. Combinatorial Theory 9 (1970), 174β191. MR 266771
- 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 (1960), 191β209. MR 111695, DOI 10.1090/S0002-9947-1960-0111695-3
- A. E. Brouwer, Some unitals on $28$ points and their embeddings in projective planes of order $9$, Geometries and groups (Berlin, 1981) Lecture Notes in Math., vol. 893, Springer, Berlin-New York, 1981, pp.Β 183β188. MR 655065
- J. H. Conway and Vera Pless, On the enumeration of self-dual codes, J. Combin. Theory Ser. A 28 (1980), no.Β 1, 26β53. MR 558873, DOI 10.1016/0097-3165(80)90057-6
- Philippe Delsarte, A geometric approach to a class of cyclic codes, J. Combinatorial Theory 6 (1969), 340β358. MR 267957 J. F. Dillon, Private communication.
- J. F. Dillon, Elementary Hadamard difference sets, Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1975) Congressus Numerantium, No. XIV, Utilitas Math., Winnipeg, Man., 1975, pp.Β 237β249. MR 0409221 J. F. Dillon and J. R. Schatz, Block designs with the symmetric difference property, (Robert L. Ward, Ed.), Proc. NSA Mathematical Sciences Meetings, The United States Government, 1987, pp. 159-164.
- Jean Doyen, Xavier Hubaut, and Monique Vandensavel, Ranks of incidence matrices of Steiner triple systems, Math. Z. 163 (1978), no.Β 3, 251β259. MR 513730, DOI 10.1007/BF01174898
- J.-M. Goethals and J. J. Seidel, Strongly regular graphs derived from combinatorial designs, Canadian J. Math. 22 (1970), 597β614. MR 282872, DOI 10.4153/CJM-1970-067-9
- Ken Gray, Further results on designs carried by a code, Ars Combin. 26 (1988), no.Β B, 133β152. MR 990544
- Marshall Hall Jr., Combinatorial theory, 2nd ed., Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, Inc., New York, 1986. A Wiley-Interscience Publication. MR 840216
- N. Hamada and H. Ohmori, On the BIB design having the minimum $p$-rank, J. Combinatorial Theory Ser. A 18 (1975), 131β140. MR 416939, DOI 10.1016/0097-3165(75)90001-1
- J. W. P. Hirschfeld, Projective geometries over finite fields, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1979. MR 554919
- D. R. Hughes and F. C. Piper, Design theory, Cambridge University Press, Cambridge, 1985. MR 812053, DOI 10.1017/CBO9780511566066
- Noboru Ito, Jeffrey S. Leon, and Judith Q. Longyear, Classification of $3-(24,\,12,\,5)$ designs and $24$-dimensional Hadamard matrices, J. Combin. Theory Ser. A 31 (1981), no.Β 1, 66β93. MR 626442, DOI 10.1016/0097-3165(81)90054-6
- Dieter Jungnickel and Vladimir D. Tonchev, On symmetric and quasi-symmetric designs with the symmetric difference property and their codes, J. Combin. Theory Ser. A 59 (1992), no.Β 1, 40β50. MR 1141321, DOI 10.1016/0097-3165(92)90097-E
- J. D. Key and K. Mackenzie, Ovals in the designs $W(2^m)$, Ars Combin. 33 (1992), 113β117. MR 1174835
- Hiroshi Kimura, Classification of Hadamard matrices of order $28$ with Hall sets, Discrete Math. 128 (1994), no.Β 1-3, 257β268. MR 1271869, DOI 10.1016/0012-365X(94)90117-1
- Hiroshi Kimura, On equivalence of Hadamard matrices, Hokkaido Math. J. 17 (1988), no.Β 1, 139β146. MR 928471, DOI 10.14492/hokmj/1381517792
- Hiroshi Kimura, New Hadamard matrix of order $24$, Graphs Combin. 5 (1989), no.Β 3, 235β242. MR 1027704, DOI 10.1007/BF01788676
- Eric S. Lander, Symmetric designs: an algebraic approach, London Mathematical Society Lecture Note Series, vol. 74, Cambridge University Press, Cambridge, 1983. MR 697566, DOI 10.1017/CBO9780511662164
- Jeffrey S. Leon, Vera Pless, and N. J. A. Sloane, On ternary self-dual codes of length $24$, IEEE Trans. Inform. Theory 27 (1981), no.Β 2, 176β180. MR 633414, DOI 10.1109/TIT.1981.1056328
- J. S. Leon, V. Pless, and N. J. A. Sloane, Self-dual codes over $\textrm {GF}(5)$, J. Combin. Theory Ser. A 32 (1982), no.Β 2, 178β194. MR 654620, DOI 10.1016/0097-3165(82)90019-X K. Mackenzie, Codes of designs, Ph.D. thesis, Univ. of Birmingham, 1989. F. J. MacWilliams and N. J. A. Sloane, The theory of error-correcting codes, North-Holland, 1983.
- F. J. MacWilliams, N. J. A. Sloane, and J. G. Thompson, Good self dual codes exist, Discrete Math. 3 (1972), 153β162. MR 307799, DOI 10.1016/0012-365X(72)90030-1
- Henry B. Mann, Addition theorems: The addition theorems of group theory and number theory, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1965. MR 0181626
- Antonio Maschietti, Hyperovals and Hadamard designs, J. Geom. 44 (1992), no.Β 1-2, 107β116. MR 1169413, DOI 10.1007/BF01228287
- C. W. Norman, Nonisomorphic Hadamard designs, J. Combinatorial Theory Ser. A 21 (1976), no.Β 3, 336β344. MR 419258, DOI 10.1016/0097-3165(76)90006-6
- Vera Pless and N. J. A. Sloane, Binary self-dual codes of length $24$, Bull. Amer. Math. Soc. 80 (1974), 1173β1178. MR 689175, DOI 10.1090/S0002-9904-1974-13662-1
- Chester J. Salwach, Planes, biplanes, and their codes, Amer. Math. Monthly 88 (1981), no.Β 2, 106β125. MR 606250, DOI 10.2307/2321134
- S. S. Shrikhande and N. K. Singh, On a method of constructing symmetrical balanced incomplete block designs, SankhyΔ Ser. A 24 (1962), 25β32. MR 144435 J. A. Todd, A combinatorial problem, J. Math. Phys. 12 (1933), 321-333.
- Vladimir D. Tonchev, Hadamard matrices of order $28$ with automorphisms of order $7$, J. Combin. Theory Ser. A 40 (1985), no.Β 1, 62β81. MR 804869, DOI 10.1016/0097-3165(85)90047-0
- Michael A. Wertheimer, Oval designs in quadrics, Finite geometries and combinatorial designs (Lincoln, NE, 1987) Contemp. Math., vol. 111, Amer. Math. Soc., Providence, RI, 1990, pp.Β 287β297. MR 1079752, DOI 10.1090/conm/111/1079752 β, Designs in quadrics, Ph.D. thesis, Univ. of Pennsylvania, 1986.
Bibliographic Information
- © Copyright 1992 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 330 (1992), 257-268
- MSC: Primary 58F05; Secondary 22E46
- DOI: https://doi.org/10.1090/S0002-9947-1992-1040046-6
- MathSciNet review: 1040046