B. Bolt, T. G. Room and G. E. Wall, On Cli#ord collineation, transform and similarity groups II, J. Australian Math. Soc. 2 (1961), 80--96.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....has order 5160960. Shortly afterwards, the same 8 dimensional group arose in the work of P. W. Shor and others on quantum computers (cf. 2] 18] This astonishing coincidence see [11] for the full story drew attention to earlier work on the family to which this group belongs [5] 6] [7], 8] 32] Following Wall, we call these Cli#ord groups, although these are not the groups usually referred to by this name [22] Investigation of the representations of subgroups of these groups led to further constructions of optimal packings in Grassmann manifolds [9] and constructions of ....

B. Bolt, T. G. Room and G. E. Wall, On Cli#ord collineation, transform and similarity groups II, J. Australian Math. Soc. 2 (1961), 80--96.

....In 1959 Barnes and Wall [2] constructed a family of lattices in dimensions 2 m , m = 0; 1; 2; They distinguish two geometrically similar lattices Lm L 0 m in R 2 m . The automorphism group y Gm = Aut(Lm ) was investigated in a series of papers by Bolt, Room and Wall [8] 9] [10], 48] Gm is a subgroup of index 2 in a certain group Cm of structure 2 1 2m :O (2m; 2) We follow Bolt et al. in calling Cm a Cli ord group. This group and its complex analogue CI m are the subject of the present paper. These groups have appeared in several di erent contexts in recent ....

....Z[ 8 ] lattice Z[ 8 ] Z[ p 2] Mm . Then CI m is the subgroup of U(2 m ; Q [ 8 ] preserving M m (Proposition 6.4) Theorem 6.5 shows that, apart from the center, CI m is a maximal nite subgroup of U(2 m ; CI ) and Corollary 6.6 is the analogue of Corollary 5.7. Bolt et al. 8] 9] [10], 48] and Sidelnikov [42] 43] 44] also consider the group C (p) m obtained by replacing 2 in the de nition of Cm by an odd prime p. In the nal section we give some analogous results for this group. In recent years many other kinds of self dual codes have been studied by a number of ....

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B. Bolt, T. G. Room and G. E. Wall, On Cliord collineation, transform and similarity groups II, J. Australian Math. Soc. 2 (1961), 80-96. 21

....In 1959 Barnes and Wall [2] constructed a family of lattices in dimensions 2 m , m = 0; 1; 2; They distinguish two geometrically similar lattices Lm L 0 m in R 2 m . The automorphism group y Gm = Aut(Lm ) was investigated in a series of papers by Bolt, Room and Wall [8] 9] [10], 48] Gm is a subgroup of index 2 in a certain group Cm of structure 2 1 2m :O (2m; 2) We follow Bolt et al. in calling Cm a Clifford group. This group and its complex analogue CI m are the subject of the present paper. These groups have appeared in several different contexts in recent ....

....8 ] lattice Z[i 8 ] Omega Z[ p 2] Mm . Then CI m is the subgroup of U(2 m ; Q[i 8 ] preserving M m (Proposition 6.4) Theorem 6.5 shows that, apart from the center, CI m is a maximal finite subgroup of U(2 m ; CI) and Corollary 6.6 is the analogue of Corollary 5.7. Bolt et al. 8] 9] [10], 48] and Sidelnikov [42] 43] 44] also consider the group C (p) m obtained by replacing 2 in the definition of Cm by an odd prime p. In the final section we give some analogous results for this group. In recent years many other kinds of self dual codes have been studied by a number of ....

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B. Bolt, T. G. Room and G. E. Wall, On Clifford collineation, transform and similarity groups II, J. Australian Math. Soc. 2 (1961), 80--96. 21

....to be identical (not just isomorphic) We then discovered that this group was a member of an infinite family of groups that played a central role in a joint paper [9] written by another colleague, A. R. Calderbank. This is a certain family of Clifford groups (the name is due to Wall [4] [5], 41] which may be constructed as follows. The starting point is the standard method of associating a finite orthogonal space to an extraspecial 2 group, as described for example in [1] Theorem 23.10, or [28] Theorem 13.8. The end result will be the construction of various packings of ....

....the center, and so also acts on E. In fact L acts on E as the orthogonal group O (2i; 2) 9] Lemma 2. 14) This Clifford group L has arisen in several different contexts, providing a link between the the problem of packing in Grassmannian spaces, the Barnes Wall lattices (see [4] [5], 37] 41] the construction of orthogonal spreads and Kerdock sets [9] and the construction of quantum 3 [28] p. 349 error correcting codes [3] 12] It also occurs in several purely group theoretic contexts see [9] for references. The connection with quantum computing arises because ....

B. Bolt, T. G. Room and G. E. Wall, On Clifford collineation, transform and similarity groups II, J. Australian Math. Soc., 2 (1961), 80--96.

....is the normalizer of ER in the orthogonal group O(2 n ) The group ER consists of the tensor products Sigmaw 1 Omega Delta Delta Delta Omega w n , where each w j is one of I , oe x , oe z , oe x oe z . ER is an extraspecial 2 group with order 2 2n 1 and 2 We follow Bolt et al. 6] [7]) in calling these Clifford groups. The same name is used for a different family of groups by Chevalley [19] and Jacobson [42] center f SigmaI g, and ER =f SigmaI g = E= Xi(E) E. For many applications it is simpler to work with the real groups ER and LR rather than E and L. The following ....

.... (2 n Gamma 1) n Gamma1 Y j=1 (4 j Gamma 1) ffl L acts on E as the symplectic group Sp 2n (2) and LR acts on E as the orthogonal group O 2n (2) The groups L and LR have arisen in several different contexts, and provide a link between quantum codes, the Barnes Wall lattices [6] [7], 76] the construction of orthogonal spreads and Kerdock sets [12] the construction of spherical codes [43] 69] 70] and the construction of Grassmannian packings [68] 13] They have also occurred in several purely group theoretic contexts see [12] for references. These groups are ....

B. Bolt, T. G. Room and G. E. Wall, On Clifford collineation, transform and similarity groups II, J. Australian Math. Soc., 2 (1961), 80--96.

....on the subspaces. In the rest of this section we shall therefore give only a brief discussion of these groups, in order to show their connection with the Barnes Wall lattices. It turns out that H i and G i are well known groups. H i is the Clifford group CT 1 (2 i ) studied in [4] [5], 14] which in recent years has been used in the classification of finite simple groups (see the references in [9] H i is relevant for the present work because of its connection with the Barnes Wall lattices. Although the original paper of Barnes and Wall [3] describes a family of lattices in ....

B. Bolt, T. G. Room and G. E. Wall, On Clifford collineation, transform and similarity groups II, J. Australian Math. Soc., 2 (1961), 80--96.

....L acts on E by conjugation, fixing the center, and so also acts on E. In fact L acts on E as the orthogonal group O (2i; 2) 7] Lemma 2. 14) This Clifford group L has arisen in several different contexts, providing a link between the present problem, the Barnes Wall lattices (see [4] [5], 20] 22] the construction of orthogonal spreads and Kerdock sets [7] and the construction of quantum error correcting codes [3] 9] It also occurs in several purely group theoretic contexts see [7] for references. 3 [15] p. 349 The connection with quantum computing arises because ....

B. Bolt, T. G. Room and G. E. Wall, On Clifford collineation, transform and similarity groups II, J. Australian Math. Soc., 2 (1961), 80--96.

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