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-

Efficacy of plasma-polymerized allylamine coating of zirconia after five years

Plasma-polymerized allylamine (PPAAm) coatings of titanium enhance the cell behavior of osteoblasts. The purpose of the present study was to evaluate a PPAAm nanolayer on zirconia after a storage period of 5 years. Zirconia specimens were directly coated with PPAAm (ZA0) or stored in aseptic packages at room temperature for 5 years (ZA5). Uncoated zirconia specimens (Zmt) and the micro-structured endosseous surface of a zirconia implant (Z14) served as controls. The elemental compositions of the PPAAm coatings were characterized and the viability, spreading and gene expression of human osteoblastic cells (MG-63) were assessed. The presence of amino groups in the PPAAm layer was significantly decreased after 5 years due to oxidation processes. Cell viability after 24 h was significantly higher on uncoated specimens (Zmt) than on all other surfaces. Cell spreading after 20 min was significantly higher for Zmt = ZA0 > ZA5 > Z14, while, after 24 h, spreading also varied significantly between Zmt > ZA0 > ZA5 > Z14. The expression of the mRNA differentiation markers collagen I and osteocalcin was upregulated on untreated surfaces Z14 and Zmt when compared to the PPAAm specimens. Due to the high biocompatibility of zirconia itself, a PPAAm coating may not additionally improve cell behavior