The Invariants of the Clifford Groups

Designs, Codes, and Cryptography

Volumn 24, Issue 1, 2001, Pages 99-122

  Nebe, Gabriele ^a   Rains, E M ^b   Sloane, N J A ^b  

^a UNIVERSITY OF ULM   (Germany)

^b AT AND T LABS RESEARCH   (United States)

Author keywords

Barnes Wall lattices; Clifford groups; Invariants; Self dual codes; Spherical designs

Indexed keywords

EID: 0035452545     PISSN: 09251022     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1011233615437     Document Type: Article

References (56)

1. E. Bannai, S. T. Dougherty, M. Harada and M. Oura, Type II codes, even unimodular lattices and invariant rings, IEEE Trans. Inform. Theory, Vol. 45 (1999) pp. 1194-1205.
2. E. S. Barnes and G. E. Wall, Some extreme forms defined in terms of Abelian groups, J. Australian Math. Soc., Vol. 1 (1959) pp. 47-63.
3. H.-J. Bartels, Zur Galoiskohomologie definiter arithmetischer Gruppen, J. Reine Angew. Math., Vol. 298 (1978) pp. 89-97.
4. C. H. Bennett, D. DiVincenzo, J. A. Smolin and W. K. Wootters, Mixed state entanglement and quantum error correction, Phys. Rev. A, Vol. 54 (1996) pp. 3824-3851.
5. D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge University Press (1993).
6. H. F. Blichfeldt, Finite Collineation Groups, University of Chicago Press, Chicago (1917).
7. S. Böcherer, Siegel modular forms and theta series, in Theta functions (Bowdoin 1987), Proc. Sympos. Pure Math., 49, Part 2, Amer. Math. Soc., Providence, RI, (1989), pp. 3-17.
8. B. Bolt, The Clifford collineation, transform and similarity groups III: Generators and involutions, J. Australian Math. Soc., Vol. 2 (1961) pp. 334-344.
9. B. Bolt, T. G. Room and G. E. Wall, On Clifford collineation, transform and similarity groups I, J. Australian Math. Soc., Vol. 2 (1961) pp. 60-79.
10. B. Bolt, T. G. Room and G. E. Wall, On Clifford collineation, transform and similarity groups II, J. Australian Math. Soc., Vol. 2 (1961) pp. 80-96.
11. R. Brauer, Über endliche lineare Gruppen von Primzahlgrad, Math. Annalen, Vol. 169 (1967) pp. 73-96
12. e série, 6 (1973), 17-52. Summary in C. R. Acad. Sc. Paris, Vol. 274 (1972) pp. 19-22.
13. e série, 6 (1973), 17-52. Summary in C. R. Acad. Sc. Paris, Vol. 274 (1972) pp. 19-22.
14. 4-Kerdock codes, orthogonal spreads and extremal Euclidean line-sets, Proc. London Math. Soc., Vol. 75 (1997) pp. 436-480.
15. A. R. Calderbank, R. H. Hardin, E. M. Rains, P. W. Shor and N. J. A. Sloane, A group-theoretic framework for the construction of packings in Grassmannian spaces, J. Algebraic Combin., Vol. 9 (1999) pp. 129-140.
16. A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction orthogonal geometry, Phys. Rev. Letters, Vol. 78 (1997) pp. 405-409.
17. A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over G F(4), IEEE Trans. Inform. Theory, Vol. 44 (1998) pp. 1369-1387.
18. J. H. Conway, R. H. Hardin and N. J. A. Sloane, Packing lines, planes, etc.: packings in Grassmannian space, Experimental Math., Vol. 5 (1996) pp. 139-159.
19. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed., Springer-Verlag, New York (1998).
20. A. W. Dress, Induction and structure theorems for orthogonal representations of finite groups, Annals of Mathematics, Vol. 102 (1975) pp. 291-325.
21. W. Duke, On codes and Siegel modular forms, Intern. Math. Res. Notices, Vol. 5 (1993) pp. 125-136.
22. W. Feit, On finite linear groups in dimension at most 10, in Proceedings of the Conference on Finite Groups (Univ. Utah, Park City, Utah, 1975), Academic Press, New York (1976) pp. 397-407.
23. A. Fröhlich and M. J. Taylor, Algebraic Number Theory, Cambridge Univ. Press (1991).
24. n, of certain extensions of 2-groups by orthogonal and symplectic groups. J. Australian Math. Soc. Ser. A, Vol. 58 (1995) pp. 232-247.
25. A. M. Gleason, Weight polynomials of self-dual codes and the MacWilliams identities, in Actes, Congrés International de Mathématiques (Nice, 1970), Gauthiers-Villars, Paris, Vol. 3 (1971) pp. 211-215.
26. J.-M. Goethals and J. J. Seidel, The football, Nieuw Archief voor Wiskunde, Vol. 29 (1981) pp. 50-58. Reprinted in Geometry and Combinatorics: Selected Works of J. J. Seidel, (D. G. Corneil and R. Mathon, ed.) Academic Press (1991) pp. 363-371.
27. J.-M. Goethals and J. J. Seidel, The football, Nieuw Archief voor Wiskunde, Vol. 29 (1981) pp. 50-58. Reprinted in Geometry and Combinatorics: Selected Works of J. J. Seidel, (D. G. Corneil and R. Mathon, ed.) Academic Press (1991) pp. 363-371.
28. W. C. Huffman and D. B. Wales, Linear groups of degree nine with no elements of order seven. J. Algebra, Vol. 51 (1978) pp. 149-163.
29. B. Huppert, Endliche Gruppen I, Springer-Verlag (1967).
30. L. S. Kazarin, On certain groups defined by Sidelnikov (in Russian), Mat. Sb., Vol. 189 (No. 7, 1998) pp. 131-144; English translation in Sb. Math., Vol. 189 (1998) pp. 1087-1100.
31. L. S. Kazarin, On certain groups defined by Sidelnikov (in Russian), Mat. Sb., Vol. 189 (No. 7, 1998) pp. 131-144; English translation in Sb. Math., Vol. 189 (1998) pp. 1087-1100.
32. A. Y. Kitaev, Quantum computations: algorithms and error correction (in Russian), Uspekhi Mat. Nauk., Vol. 52 (No. 6, 1997) pp. 53-112; English translation in Russian Math. Surveys, Vol. 52 (1997) pp. 1191-1249.
33. A. Y. Kitaev, Quantum computations: algorithms and error correction (in Russian), Uspekhi Mat. Nauk., Vol. 52 (No. 6, 1997) pp. 53-112; English translation in Russian Math. Surveys, Vol. 52 (1997) pp. 1191-1249.
34. P. B. Kleidman and M. W. Liebeck, The Subgroup Structure of the Finite Classical Groups, Cambridge Univ. Press (1988).
35. J. H. Lindsey II, Finite linear groups of prime degree, Math. Annalen, Vol. 189 (1970) pp. 47-59.
36. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes North-Holland, Amsterdam (1977).
37. G. Nebe, Finite quaternionic matrix groups, Representation Theory, Vol. 2 (1998) pp. 106-223.
38. G. Nebe, E. M. Rains and N. J. A. Sloane, A simple construction for the Barnes-Wall lattices, in Forney Festschrift, (R. Blahut, eds.) to appear (2001).
39. G. Nebe, E. M. Rains and N. J. A. Sloane, Generalized self-dual codes and Clifford-Weil groups, preprint.
40. M. Oura, The dimension formula for the ring of code polynomials in genus 4, Osaka J. Math., Vol. 34 (1997) pp. 53-72.
41. V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Academic Press, San Diego (1994).
42. E. M. Rains and N. J. A. Sloane, Self-dual codes, in Handbook of Coding Theory, ed. V. Pless and W. C. Huffman, Elsevier, Amsterdam (1998) pp. 177-294.
43. B. Runge, On Siegel modular forms I, J. Reine Angew. Math., Vol. 436 (1993) pp. 57-85.
44. B. Runge, On Siegel modular forms II, Nagoya Math. J., Vol. 138 (1995) pp. 179-197.
45. B. Runge, The Schottky ideal, in Abelian Varieties (Egloffstein, 1993), de Gruyter, Berlin (1995) pp. 251-272.
46. B. Runge, Codes and Siegel modular forms, Discrete Math., Vol. 148 (1996) pp. 175-204.
47. J.-P. Serre, Linear Representations of Finite Groups, Springer-Verlag (1977).
48. P. W. Shor and N. J. A. Sloane, A family of optimal packings in Grassmannian manifolds, J. Algebraic Combin., Vol. 7 (1998) pp. 157-163.
49. V. M. Sidelnikov, On a finite group of matrices and codes on the Euclidean sphere (in Russian), Probl. Peredach. Inform., (1997), Vol. 33 pp. 33 35-54 (1997); English translation in Problems Inform. Transmission, Vol. 33 (1997) pp. 29-44.
50. V. M. Sidelnikov, On a finite group of matrices and codes on the Euclidean sphere (in Russian), Probl. Peredach. Inform., (1997), Vol. 33 pp. 33 35-54 (1997); English translation in Problems Inform. Transmission, Vol. 33 (1997) pp. 29-44.
51. V. M. Sidelnikov, On a finite group of matrices generating orbit codes on the Euclidean sphere, in Proceedings IEEE Internat. Sympos. Inform. Theory, Ulm, 1997, IEEE Press (1997) p. 436.
52. n-dimensional Euclidean space, J. Algebraic Combin., Vol. 10 (1999) pp. 279-288.
53. V. M. Sidelnikov, Orbital spherical 11-designs in which the initial point is a root of an invariant polynomial (in Russian), Algebra i Analiz, Vol. 11 (No. 4, 1999) pp. 183-203.
54. B. Venkov, Réseaux et "designs" sphériques, in Réseaux euclidiens, "designs" sphériques et groupes, L'Enseignement Mathématiques Monographie Vol. 37 (Martinet, J. ed.), to appear (2001).
55. G. E. Wall, On Clifford collineation, transform and similarity groups IV: an application to quadratic forms, Nagoya Math. J., Vol. 21 (1962) pp. 199-222.
56. D. L. Winter, The automorphism group of an extraspecial p-group, Rocky Mtn. J. Math., Vol. 2 (1972) pp. 159-168.