The invariants of the Clifford groups

The automorphism group of the Barnes-Wall lattice L_m in dimension 2^m (m not 3) is a subgroup of index 2 in a certain ``Clifford group'' C_m (an extraspecial group of order 2^(1+2m) extended by an orthogonal group). This group and its complex analogue CC_m have arisen in recent years in connection with the construction of orthogonal spreads, Kerdock sets, packings in Grassmannian spaces, quantum codes, Siegel modular forms and spherical designs. In this paper we give a simpler proof of Runge's 1996 result that the space of invariants for C_m of degree 2k is spanned by the complete weight enumerators of the codes obtained by tensoring binary self-dual codes of length 2k with the field GF(2^m); these are a basis if m >= k-1. We also give new constructions for L_m and C_m: let M be the Z[sqrt(2)]-lattice with Gram matrix [2, sqrt(2); sqrt(2), 2]. Then L_m is the rational part of the mth tensor power of M, and C_m is the automorphism group of this tensor power. Also, if C is a binary self-dual code not generated by vectors of weight 2, then C_m is precisely the automorphism group of the complete weight enumerator of the tensor product of C and GF(2^m). There are analogues of all these results for the complex group CC_m, with ``doubly-even self-dual code'' instead of ``self-dual code''.Comment: Latex, 24 pages. Many small improvement

Spherical designs and lattices

In this article we prove that integral lattices with minimum <= 7 (or <= 9) whose set of minimal vectors form spherical 9-designs (or 11-designs respectively) are extremal, even and unimodular. We furthermore show that there does not exist an integral lattice with minimum <=11 which yields a 13-design.Comment: The final publication is available at http://link.springer.com/article/10.1007%2Fs13366-013-0155-

Restricted cell functions on micropillars are alleviated by surface-nanocoating with amino groups

The topographical and chemical surface features of biomaterials are sensed by the cells, affecting their physiology at the interface. When placed on titanium, we recently discovered osteoblasts attempted caveolae-mediated phagocytosis of the sharp-edged microstructures. This active, energy-consuming process resulted in decreased osteoblastic cell functions (e.g. secretion of extracellular matrix proteins). However, chemical modification with plasma polymerized allylamine (PPAAm) was able to amplify osteoblast adhesion and spreading, resulting in better implant osseointegration in vivo. In the present in vitro study, we analyzed whether this plasma polymer nanocoating is able to attenuate the microtopography-induced changes of osteoblast physiology. On PPAAm, we found cells showed a higher cell interaction with the geometrical micropillars by 30 min, and a less distinct reduction in the mRNA expression of collagen type I, osteocalcin and fibronectin after 24 h of cell growth. Interestingly, the cells were more active and sensitive on PPAAm-coated micropillars, and react with a substantial Ca2+ ion mobilization after stimulation with ATP. These results highlight that it is important for osteoblasts to establish cell surface contact for them to perform their functions