有限域上典型群的BN-对及有限T-群的分类
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摘要
令G是群,子群B和N是G的BN-对,T=B∩N是N的正规子群,则商群W=N/T是一个有限反射群,设W的极小生成集为S,那么∑=∑(W,S)是一个Building.本文的目的是研究典型群中非经典BN-对的构造,以及BN-对的有理不变式域.另外,本文还研究了Z_2上不可约有限T-群的分类问题.
第一章用矩阵的方法构造了有限域上典型群的非经典的BN-对,并给出了子群B和N的有理不变式域的生成元集.
第二章找出了有限域上(广义)典型群的子群B,N以及T的有理不变式域的完整的代数无关的生成元集,并计算子了群B,N以及T的阶数.
第三章讨论了Z_2上本质的不可约有限T-群的分类问题.我们利用T-群与Cartan矩阵的关系,在群同构意义下重新给出了群的生成元,并计算了群的阶数.
第四章利用特征数为2的有限域上辛对合矩阵构造了一类Catersian认证码,计算了该码的所有参数.在假定信源和编码规则按照等概率均匀分布的条件下,给出了该认证码被成功模仿攻击的最大概率P_I和被成功替换攻击的最大概率P_S.
Let G be a group, B and N be the BN-pairs of G, and T = B∩N is a normal subgroup of TV, then we get a finite reflection group W = N/T. Let S be a minimal generating set of W, thenΣ=Σ(W, S) is a Building. The purpose of this thesis is to study the non-classical construction of BN-pairs of classical groups, the rational invariants of subgroups B and N, and also we consider the classification of the finite irreducible T-groups over Z_2.
In Chapter 1, we construct non-classical BN-pairs arising from the classical groups over finite fields. Furthermore, we give transcendence bases of the rational invariant field of subgroups B and N.
In Chapter 2, we construct explicit transcendence bases of the rational invariant fields of the generalized classical groups and subgroups B, N and T, and we also compute the orders of them.
In Chapter 3, we determine the structure of the finite effective irreducible T-groups over Z_2. Up to isomorphism, we give explicit generators of the T-groups by using the Cartan matrices, and we also compute the orders of them.
In Chapter 4, we compute the number of the symplectic involutions over the finite field F with char F = 2, and also one Cartesian authentication code is obtained. Furthermore, its size parameters are computed completely. Assume that the coding rules are chosen according to a uniform probability, P_I and P_S denote the largest probabilities of a successful impersonation attack and a successful substitution attack respectively, then P_I and P_S are also computed.
第一章用矩阵的方法构造了有限域上典型群的非经典的BN-对,并给出了子群B和N的有理不变式域的生成元集.
第二章找出了有限域上(广义)典型群的子群B,N以及T的有理不变式域的完整的代数无关的生成元集,并计算子了群B,N以及T的阶数.
第三章讨论了Z_2上本质的不可约有限T-群的分类问题.我们利用T-群与Cartan矩阵的关系,在群同构意义下重新给出了群的生成元,并计算了群的阶数.
第四章利用特征数为2的有限域上辛对合矩阵构造了一类Catersian认证码,计算了该码的所有参数.在假定信源和编码规则按照等概率均匀分布的条件下,给出了该认证码被成功模仿攻击的最大概率P_I和被成功替换攻击的最大概率P_S.
Let G be a group, B and N be the BN-pairs of G, and T = B∩N is a normal subgroup of TV, then we get a finite reflection group W = N/T. Let S be a minimal generating set of W, thenΣ=Σ(W, S) is a Building. The purpose of this thesis is to study the non-classical construction of BN-pairs of classical groups, the rational invariants of subgroups B and N, and also we consider the classification of the finite irreducible T-groups over Z_2.
In Chapter 1, we construct non-classical BN-pairs arising from the classical groups over finite fields. Furthermore, we give transcendence bases of the rational invariant field of subgroups B and N.
In Chapter 2, we construct explicit transcendence bases of the rational invariant fields of the generalized classical groups and subgroups B, N and T, and we also compute the orders of them.
In Chapter 3, we determine the structure of the finite effective irreducible T-groups over Z_2. Up to isomorphism, we give explicit generators of the T-groups by using the Cartan matrices, and we also compute the orders of them.
In Chapter 4, we compute the number of the symplectic involutions over the finite field F with char F = 2, and also one Cartesian authentication code is obtained. Furthermore, its size parameters are computed completely. Assume that the coding rules are chosen according to a uniform probability, P_I and P_S denote the largest probabilities of a successful impersonation attack and a successful substitution attack respectively, then P_I and P_S are also computed.
引文
[1] BEDARD R. The lowest two-sided cell for an affine Weyl group [J]. Comm. Alg., 1988, 16:1113-1132.
[2] LUSZTIG G. Cells in affine Weyl groups [J], Ⅱ: J. Algebra, 1987, 109 : 536-548.
[3] LUSZTIG G. Cells in affine Weyl groups [J], Ⅲ: J. Fac. Sci. Univ. Tokyo Math., 1987, 34:223-234.
[4] LUSZTIG G. Cells in affine Weyl groups [J], Ⅳ: J. Fac. Sci. Univ. Tokyo Math., 1989, 36:297-328.
[5] CURTIS C, REINER I. Representation Theory of Finite Groups and Associative Algebras [M]. London: John Wilery and Sons, 1962.
[6] BOROVIK A, NESIN A. Groups of finite Morley rank [M]. New York: Oxford University Press, 1994.
[7] COOPERSTEIN B N. Geometry of groups of Lie type [M]. New York: Pro. Conf. Finite Groups Utah, Academic Press, 1976.
[8] GORENSTEIN D. Finite Simple Groups [M]. New York: Plenum Press, 1982.
[9] CARTER R W. Simple Groups of Lie Type [M]. New York: John Wiley and Sons, 1972.
[10] HUMPHREYS J E. Introduction to Lie algebras and Representation Theory [Ml. New York: Springer-Verlag, 1972.
[11] ASCHBACHER M. On finite groups of Lie type and odd characteristic [J]. Algebra, 1980,66: 400-424.
[12] BJORNER A. Some combinatorial and algerbraic properties of Coxeter complex and Tits building [J]. Adv. in Math., 1984, 52: 173-212.
[13] DEODHAR V V. On some geometric aspects of Bruhat orderings [J], I: A finer decomposition of Bruhat cells. Inv. Math., 1985, 79: 499-511.
[14] FONG P, SEITZ G. Groups with a BN-pair of rank 2 [J], I: Invent. Math., 1973, 21: 1- 57.
[15] FONG P, SEITZ G. Groups with a BN-pair of rank 2 [J], II: Invent. Math., 1974, 24:191 - 239.
[16] DEODHAR V V. On some geometric aspects of Bruhat orderings [J], Ⅱ: The parabolic analogue of Kazhdan-Luszing polynomials. J. Algebra, 1987, 111: 483-506.
[17] TENT K. Split BN-pairs of rank 2: the octagons [J]. Adv. in Mathematics, 2004, 181:308 - 320.
[18] TITS J. Building of Spherical type and finite BN-pairs [M]. 1st ed. Volume 386. Berlin:Springer, 1974.
[19] HILLER H. Geometry of Coxeter Groups [M]. Volume 54. Boston: Pitman Advanced Publishing Program, 1982.
[20] BROWN K S. Buildings [M]. New York: Springer-Verlag, 1989.
[21] GARRETT P. Buildings and Classical Groups [M]. London: Chapman and Hall, 1977.
[22] IWAHORI N, MATSUMOTO H. On some bruhat decomposition and the structure of the hecke ring of p-adic chevalley groups [J]. Inst. Hautes Etudes Sci. Publ. Math., 1965, 25:5-48.
[23] HUNGERFORD T W. Algebra [M]. New York: Springer-Verlag, 1974.
[24] SWAN R G. Invariants rational functions and a problem of Steenord [J]. Invent. Math..1969, 7: 148-158.
[25] SALTMAN D J. Noether's problem over an algebraically closed field [J]. Invent. Math..1984, 77:71-84.
[26] DICKSON L E. A fundamental system of invariants of the general modular linear group with a solution of the form problem [J]. Trans.Amer.Math.Soc, 1911, 12: 75-98.
[27] CARLISLE D, KROPHOLLER P H. Rational invariants of certain orthogonal and unitary groups[J]. Bull.London Math.Soc, 1992, 24: 57-60.
[28] CHU H. Supplementary notes on "rational invariants of certain orthogonal and unitary groups" [J]. Bull.London Math.Soc, 1997, 29: 37-42.
[29] TANG Z M, WAN Z X. A matrix approach to the rational invariants of certain classical groups over finite fields of characteristic two [J]. Finite Fields and Their Applications, 2006,12: 186-210.
[30] NAGATA M. On the 14th problem of Hilbert [J]. Amer.Journal of Math., 1959, 81:766-772.
[31] ENDO S, MIYATA T. Invariants of finite abelian groups [J]. J. Math. Soc. Japan,1973,25: 7-26.
[32] RICHMAN D R. Explicit generators of the invariants of finite groups [J]. Adv. in Math., 1996, 124: 49-76.
[33] RAJAEI S M. Rational invariants of certain orthogonal groups over finite fields of characteristic two [J]. Comm. Algebra, 2000, 28: 2367-2393.
[34] MIYATA T. Invatiants of certain groups [J], I: Nagoya Math. J, 1971, 41: 69-73.
[35] KANG M. Fixed fields of triangular matrix groups [J]. J. Algebra, 2006, 302: 845-847.
[36] KANG M. Noether's problem for metacyclic p-groups [J]. Adv. in Math., 2005, 203:554-567.
[37] KANG M. Rationality problem of GL_4 group actions [J]. Adv. in Math, 2004, 181:321-352.
[38] DERKSEN H, KEMPER G. Computational invariant theory [M]. Berlin: Springer,2002.
[39] COXETER H S M. Discrete groups generated by reflcetions [J]. Canada.J.Math.,1934, 35: 588-621.
[40] SHEPHARD G C, Todd J A. Finite unitary reflection groups [J]. Canada. J. Math.,1954,6: 274-304.
[41] ZALESSKII A E, SEREZKIN V N. Finite linear groups generated by reflections [J].(Russian) Izv.Akad.Nauk SSSR Ser.Mat., 1980, 44: 1279-1307, 38. (English translation in Math.USSR Izvestija, 1981, 17: 477-503.)
[42] WAGNER A. Determination of the finite primitive reflection groups over an arbitrary field of characteristic not two. I, Geom.Dedicata 9(1980): 239-253.
[43] WAGNER A. Determination of the finite primitive reflection groups over an arbitrary field of characteristic not two. II,III, Geom.Dedicata 10(1981): 191-203, 475-523.
[44] YOU H. NAN J Z. Decomposition of Steinberg group into involutions [J]. Science in China Press, co-published with Springer-Verlag GmbH, 1997, 42(24), 2061-2064.
[45] MALDEGHAM H V. Discrete linear groups generated by reflections [J]. Math.USSR Izvestija, 1971, 5: 1083-1119.
[46] COHEN A M. Finite complex reflection groups [J]. Ann.Sci.Ecole Norm.Sup., 1976,9: 379-436.
[47] GROVE L C, BENSON C T. Finite Reflection Groups [M]. 2nd ed. Corr, Springer-Verlag New York Berlin Heidelberg Tokyo, 1971.
[48] NAN J Z, YOU H. Products of Involutions in Steinberg Group over Skew Fields [J].Chinese Annals of Mathematics-Series B, Springer Berlin/Herdelberg, 2007, 2(28):253-264.
[49] RADJAVI H. Decomposition of matrices into simple involutions [J]. Algebra and Appl., 1975, 12: 247-255.
[50] COHEN A M. Finite quaternionic reflection groups [J]. J.Algebra, 1980, 64: 293-324.
[51] STEINBERG R. Finite reflection groups [J]. Trans.Amer.Math.Soc, 1959, 91: 493-504.
[52] HUMPHREYS J E. Reflection Groups and Coxeter grouops [M]. London: Cambrige University Press, 1990.
[53] MCLAUGHLIN J E. Some groups generated by transvections [J]. Arch.Math., 1967,18: 364-368.
[54] MC1AUGHLIN J E. Some subgroups of SL_n(F_2) [J]. Illinois J.Math., 1969, 13: 108-115.
[55] POLLATSEK H. Irreducible groups generated by transvections over finite fields of characteristic two [J]. J.Algebra, 1976, 39: 328-333.
[56] ZALESSKII A E, SERE(?)KIN V N. Linear groups generated by transvections [J].(Russian) Izv.Akad.Nauk SSSR Ser.Mat., 1976, 40: 26-49. (English translation in Math.USSR Izvestija, 1976, 10: 25-46.)
[57] MONSON B, SCHULTE E. Reflection Groups and Polytopes over Finite Fields [J],I: Adv. in Applied Mathematics, 2004, 33(2): 290-317.
[58] MONSON B, SCHULTE E, Reflection Groups and Polytopes over Finite Fields [J].II: Adv. in Applied Mathematics, 2007, 38(3): 327-356.
[59] MONSON B, Schulte Egon. Reflection Groups and Polytopes over Finite Fields [J],III: Adv. in Applied Mathematics, 2008, 41(1): 76-94.
[60] SIMMONS G J. Authentication Theory/Coding Theory [J]. Adv. in Cryptology,1985, 196: 411-431.
[61] WAN Z X. Further Constructions of Cartesian authentication codes from symplectic geometry [J]. J. Northeastern Math., 1992, 8:4-20.
[62] WAN Z X. Construction of Cartesian authentication codes from unitary geometry [J]. Designs, Codes and Cryptography, 1992, 2: 333-356.
[63] YOU H, NAN J Z. Using Normal form of matrices over finite fields to construct Cartesian authentication codes [J]. Math Research and Exposition, 1998, 18(3), 341-346.
[64] SIMMONS G J. Message authentication with arbitration of transmitter/receiver disputes [A]. Proceedings of Cryto-87 [C]. LNC304, Springer Verlag, Berlin, 1987: 151-165.
[65] CARTER R W. Finite Groups of Lie Type [M]. New York: John Wiley and Sons,1985.
[66] HUA L K, WAN Z X. Classical Groups(in Chinese) [M]. Shanghai: Shanghai Science and Technology Press, 1963.
[67] WAN Z X. Geometry of Classical Groups over Finite Fields [M]. Beijing: Science Press, 2002.
[68] BENSON D J. Polynomial invariants of finite groups [M]. London: Cambridge University Press, 1993.
[69] NEUSEL. Invariant Theory [M]. New York: AMS, 2007.
[70] NEUSEL S. Invariant Theory of Finite Groups [M]. vol. 94. New York: AMS, 2000.
[71] SANTOS W F, RITTATORE A. Actions and Invariants of Algerbraic Groups 123[M].3rd ed. Boca Raton: CRC Press, 2005.
[72] CHEVALLEY C. Invariants of finite groups generated by reflections [J]. Amer. J.Math., 1955, 77: 778-782.
[73] KANE R. Fefiection Groups and Invariant Theory [M]. New York: Springer-Verlag,2001.
[74] WEYL H. The classical groups, their invariants and representations [M]. Volume 1.Princeton Univ. Press: Princeton Mathematical Series, 1946.
[75] WILKERSON C. A primer on the Dickson invariants [J]. Amer. Math. Soc. Contemp.Math. Series, 1983, 19: 421-434.
[76] STEINBERG R. Invariants of Finite Reflection Groups [J]. Canad.J.Math., 1960,121: 616-618.
[77] LENSTRA H W. Rational functions invariant under a finite abelian group [J]. Invent.Math., 1974, 25: 299-325.
[78] CHU H. Orthogonal group actions on rational function fileds [J]. Bull. Inst. Math.Acad. Sinica, 1988, 16: 115-122.
[79] COHEN S D. Rational function invariant under an orthogonal group [J]. Bull. London Math. Soc, 1990, 22: 217-221.
[80] KEMPER G. Calculating Invariant Rings of Finite Groups over Arbitrary Fields [J].J.Symbolic Comput., 1996, 21: 351-366.
[81] NAN J Z, CHEN Y. Rational invariants of certain classical similitude groups over finite fields. Indiana Uni. Math. J., 2008, 57(4): 1947-1958.
[82] LI S Z. Subgroup Structure of Classical Groups(in Chinese) [M]. Shanghai: Shanghai Science and Technology Press, 1998.
[83] LIDL R, NIEDERREITER H. Finite Fields [M]. London: Cambridge University Press, 1983.
[84] CHERLIN G. Good tori in groups of finite Morley rank [J]. J. Group Theory, 2005,8(5): 613-622.
[85] HUA L K. Introduction to number theory (in Chinese) [M]. Beijing: Science Press,1956.
[86] WAN Z, DAI Z, FENG X, YANG B. Studies in Finite Geometry and the Constructions of Incomplete Block Designs (in Chinese) [M]. Beijing: Science Press, 1996.
[2] LUSZTIG G. Cells in affine Weyl groups [J], Ⅱ: J. Algebra, 1987, 109 : 536-548.
[3] LUSZTIG G. Cells in affine Weyl groups [J], Ⅲ: J. Fac. Sci. Univ. Tokyo Math., 1987, 34:223-234.
[4] LUSZTIG G. Cells in affine Weyl groups [J], Ⅳ: J. Fac. Sci. Univ. Tokyo Math., 1989, 36:297-328.
[5] CURTIS C, REINER I. Representation Theory of Finite Groups and Associative Algebras [M]. London: John Wilery and Sons, 1962.
[6] BOROVIK A, NESIN A. Groups of finite Morley rank [M]. New York: Oxford University Press, 1994.
[7] COOPERSTEIN B N. Geometry of groups of Lie type [M]. New York: Pro. Conf. Finite Groups Utah, Academic Press, 1976.
[8] GORENSTEIN D. Finite Simple Groups [M]. New York: Plenum Press, 1982.
[9] CARTER R W. Simple Groups of Lie Type [M]. New York: John Wiley and Sons, 1972.
[10] HUMPHREYS J E. Introduction to Lie algebras and Representation Theory [Ml. New York: Springer-Verlag, 1972.
[11] ASCHBACHER M. On finite groups of Lie type and odd characteristic [J]. Algebra, 1980,66: 400-424.
[12] BJORNER A. Some combinatorial and algerbraic properties of Coxeter complex and Tits building [J]. Adv. in Math., 1984, 52: 173-212.
[13] DEODHAR V V. On some geometric aspects of Bruhat orderings [J], I: A finer decomposition of Bruhat cells. Inv. Math., 1985, 79: 499-511.
[14] FONG P, SEITZ G. Groups with a BN-pair of rank 2 [J], I: Invent. Math., 1973, 21: 1- 57.
[15] FONG P, SEITZ G. Groups with a BN-pair of rank 2 [J], II: Invent. Math., 1974, 24:191 - 239.
[16] DEODHAR V V. On some geometric aspects of Bruhat orderings [J], Ⅱ: The parabolic analogue of Kazhdan-Luszing polynomials. J. Algebra, 1987, 111: 483-506.
[17] TENT K. Split BN-pairs of rank 2: the octagons [J]. Adv. in Mathematics, 2004, 181:308 - 320.
[18] TITS J. Building of Spherical type and finite BN-pairs [M]. 1st ed. Volume 386. Berlin:Springer, 1974.
[19] HILLER H. Geometry of Coxeter Groups [M]. Volume 54. Boston: Pitman Advanced Publishing Program, 1982.
[20] BROWN K S. Buildings [M]. New York: Springer-Verlag, 1989.
[21] GARRETT P. Buildings and Classical Groups [M]. London: Chapman and Hall, 1977.
[22] IWAHORI N, MATSUMOTO H. On some bruhat decomposition and the structure of the hecke ring of p-adic chevalley groups [J]. Inst. Hautes Etudes Sci. Publ. Math., 1965, 25:5-48.
[23] HUNGERFORD T W. Algebra [M]. New York: Springer-Verlag, 1974.
[24] SWAN R G. Invariants rational functions and a problem of Steenord [J]. Invent. Math..1969, 7: 148-158.
[25] SALTMAN D J. Noether's problem over an algebraically closed field [J]. Invent. Math..1984, 77:71-84.
[26] DICKSON L E. A fundamental system of invariants of the general modular linear group with a solution of the form problem [J]. Trans.Amer.Math.Soc, 1911, 12: 75-98.
[27] CARLISLE D, KROPHOLLER P H. Rational invariants of certain orthogonal and unitary groups[J]. Bull.London Math.Soc, 1992, 24: 57-60.
[28] CHU H. Supplementary notes on "rational invariants of certain orthogonal and unitary groups" [J]. Bull.London Math.Soc, 1997, 29: 37-42.
[29] TANG Z M, WAN Z X. A matrix approach to the rational invariants of certain classical groups over finite fields of characteristic two [J]. Finite Fields and Their Applications, 2006,12: 186-210.
[30] NAGATA M. On the 14th problem of Hilbert [J]. Amer.Journal of Math., 1959, 81:766-772.
[31] ENDO S, MIYATA T. Invariants of finite abelian groups [J]. J. Math. Soc. Japan,1973,25: 7-26.
[32] RICHMAN D R. Explicit generators of the invariants of finite groups [J]. Adv. in Math., 1996, 124: 49-76.
[33] RAJAEI S M. Rational invariants of certain orthogonal groups over finite fields of characteristic two [J]. Comm. Algebra, 2000, 28: 2367-2393.
[34] MIYATA T. Invatiants of certain groups [J], I: Nagoya Math. J, 1971, 41: 69-73.
[35] KANG M. Fixed fields of triangular matrix groups [J]. J. Algebra, 2006, 302: 845-847.
[36] KANG M. Noether's problem for metacyclic p-groups [J]. Adv. in Math., 2005, 203:554-567.
[37] KANG M. Rationality problem of GL_4 group actions [J]. Adv. in Math, 2004, 181:321-352.
[38] DERKSEN H, KEMPER G. Computational invariant theory [M]. Berlin: Springer,2002.
[39] COXETER H S M. Discrete groups generated by reflcetions [J]. Canada.J.Math.,1934, 35: 588-621.
[40] SHEPHARD G C, Todd J A. Finite unitary reflection groups [J]. Canada. J. Math.,1954,6: 274-304.
[41] ZALESSKII A E, SEREZKIN V N. Finite linear groups generated by reflections [J].(Russian) Izv.Akad.Nauk SSSR Ser.Mat., 1980, 44: 1279-1307, 38. (English translation in Math.USSR Izvestija, 1981, 17: 477-503.)
[42] WAGNER A. Determination of the finite primitive reflection groups over an arbitrary field of characteristic not two. I, Geom.Dedicata 9(1980): 239-253.
[43] WAGNER A. Determination of the finite primitive reflection groups over an arbitrary field of characteristic not two. II,III, Geom.Dedicata 10(1981): 191-203, 475-523.
[44] YOU H. NAN J Z. Decomposition of Steinberg group into involutions [J]. Science in China Press, co-published with Springer-Verlag GmbH, 1997, 42(24), 2061-2064.
[45] MALDEGHAM H V. Discrete linear groups generated by reflections [J]. Math.USSR Izvestija, 1971, 5: 1083-1119.
[46] COHEN A M. Finite complex reflection groups [J]. Ann.Sci.Ecole Norm.Sup., 1976,9: 379-436.
[47] GROVE L C, BENSON C T. Finite Reflection Groups [M]. 2nd ed. Corr, Springer-Verlag New York Berlin Heidelberg Tokyo, 1971.
[48] NAN J Z, YOU H. Products of Involutions in Steinberg Group over Skew Fields [J].Chinese Annals of Mathematics-Series B, Springer Berlin/Herdelberg, 2007, 2(28):253-264.
[49] RADJAVI H. Decomposition of matrices into simple involutions [J]. Algebra and Appl., 1975, 12: 247-255.
[50] COHEN A M. Finite quaternionic reflection groups [J]. J.Algebra, 1980, 64: 293-324.
[51] STEINBERG R. Finite reflection groups [J]. Trans.Amer.Math.Soc, 1959, 91: 493-504.
[52] HUMPHREYS J E. Reflection Groups and Coxeter grouops [M]. London: Cambrige University Press, 1990.
[53] MCLAUGHLIN J E. Some groups generated by transvections [J]. Arch.Math., 1967,18: 364-368.
[54] MC1AUGHLIN J E. Some subgroups of SL_n(F_2) [J]. Illinois J.Math., 1969, 13: 108-115.
[55] POLLATSEK H. Irreducible groups generated by transvections over finite fields of characteristic two [J]. J.Algebra, 1976, 39: 328-333.
[56] ZALESSKII A E, SERE(?)KIN V N. Linear groups generated by transvections [J].(Russian) Izv.Akad.Nauk SSSR Ser.Mat., 1976, 40: 26-49. (English translation in Math.USSR Izvestija, 1976, 10: 25-46.)
[57] MONSON B, SCHULTE E. Reflection Groups and Polytopes over Finite Fields [J],I: Adv. in Applied Mathematics, 2004, 33(2): 290-317.
[58] MONSON B, SCHULTE E, Reflection Groups and Polytopes over Finite Fields [J].II: Adv. in Applied Mathematics, 2007, 38(3): 327-356.
[59] MONSON B, Schulte Egon. Reflection Groups and Polytopes over Finite Fields [J],III: Adv. in Applied Mathematics, 2008, 41(1): 76-94.
[60] SIMMONS G J. Authentication Theory/Coding Theory [J]. Adv. in Cryptology,1985, 196: 411-431.
[61] WAN Z X. Further Constructions of Cartesian authentication codes from symplectic geometry [J]. J. Northeastern Math., 1992, 8:4-20.
[62] WAN Z X. Construction of Cartesian authentication codes from unitary geometry [J]. Designs, Codes and Cryptography, 1992, 2: 333-356.
[63] YOU H, NAN J Z. Using Normal form of matrices over finite fields to construct Cartesian authentication codes [J]. Math Research and Exposition, 1998, 18(3), 341-346.
[64] SIMMONS G J. Message authentication with arbitration of transmitter/receiver disputes [A]. Proceedings of Cryto-87 [C]. LNC304, Springer Verlag, Berlin, 1987: 151-165.
[65] CARTER R W. Finite Groups of Lie Type [M]. New York: John Wiley and Sons,1985.
[66] HUA L K, WAN Z X. Classical Groups(in Chinese) [M]. Shanghai: Shanghai Science and Technology Press, 1963.
[67] WAN Z X. Geometry of Classical Groups over Finite Fields [M]. Beijing: Science Press, 2002.
[68] BENSON D J. Polynomial invariants of finite groups [M]. London: Cambridge University Press, 1993.
[69] NEUSEL. Invariant Theory [M]. New York: AMS, 2007.
[70] NEUSEL S. Invariant Theory of Finite Groups [M]. vol. 94. New York: AMS, 2000.
[71] SANTOS W F, RITTATORE A. Actions and Invariants of Algerbraic Groups 123[M].3rd ed. Boca Raton: CRC Press, 2005.
[72] CHEVALLEY C. Invariants of finite groups generated by reflections [J]. Amer. J.Math., 1955, 77: 778-782.
[73] KANE R. Fefiection Groups and Invariant Theory [M]. New York: Springer-Verlag,2001.
[74] WEYL H. The classical groups, their invariants and representations [M]. Volume 1.Princeton Univ. Press: Princeton Mathematical Series, 1946.
[75] WILKERSON C. A primer on the Dickson invariants [J]. Amer. Math. Soc. Contemp.Math. Series, 1983, 19: 421-434.
[76] STEINBERG R. Invariants of Finite Reflection Groups [J]. Canad.J.Math., 1960,121: 616-618.
[77] LENSTRA H W. Rational functions invariant under a finite abelian group [J]. Invent.Math., 1974, 25: 299-325.
[78] CHU H. Orthogonal group actions on rational function fileds [J]. Bull. Inst. Math.Acad. Sinica, 1988, 16: 115-122.
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