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--- author: - 'Gerald Höhn[^1]' title: Conformal Designs based on Vertex Operator Algebras --- \#1 Introduction ============ In the past, it has been a fruitful approach to generalize concepts known for codes and lattices to vertex operator algebras and then to show that analogous results hold in this context. Important examples are the construction of the Moonshine module [@Bo-ur; @FLM] or the modular invariance of the genus-$1$ correlation functions [@Zhu-dr]. Other examples involving are given in [@Ho-dr; @Ho-shadow]. A comprehensive analysis of these analogies can be found in [@Ho-kleinian], where also a fourth step in this analogy, codes over the Kleinian four-group ${\bf Z}_2\times {\bf Z}_2$, was introduced. Two well-studied combinatorial structures are block designs and spherical designs and many examples of such designs are related to codes and lattices. A notion analogous to block designs and spherical designs in the context of however has been missing so far. In this paper, we introduce the notation of a [*conformal design based on a vertex operator algebra*]{} and prove several results analogous to known results for block and spherical designs. The paper is organized as follows. In the rest of the introduction we discuss the concepts of block and spherical $t$-designs to motivate our definition of conformal $t$-designs. Details of the definition will be discussed in the next section. Section 2 also contains several basic results. In particular, we prove that a vertex operator algebra $V$ with a compact automorphism group $G$ leads to conformal designs if the fixed point $V^G$ has large minimal weight (Theorem \[groupdesigns\]). The minimal weight of a $V$ is the smallest $m>0$ with $V_m\not=0$. We also describe the construction of derived designs (Theorem \[derivedparts\]). In Section 3, we prove that the homogeneous parts of an extremal are conformal $11$-, $7$-, or $3$-designs depending on the residue class $\bmod \ 24$ of the central charge of $V$. Extremal have been introduced in [@Ho-dr] and are $V$ with only one irreducible module up to isomorphism such that the first coefficients of the graded trace of $V$ are as small as possible. The Moonshine module is an example of an extremal supporting $11$-designs. This theorem is the analog of theorems of Assmus-Mattson [@AsMa] (see also [@Ba-harm]) and Venkov [@VE-design] for extremal doubly-even codes and extremal even lattices, respectively. In the final section, we study the classification of conformal $6$- and $8$-designs $V_m$ supported by $V$ of minimal weight $m\leq 2$. Our result for conformal $6$-designs (Theorem \[classfic7\]) is analogous to (but a little weaker than) similar results for codes [@LaLa-codesign] and lattices [@Ma-ladesign]: For $m=1/2$ and $m=1$, the only examples are the single Fermion $V_{\rm Fermi}$ and the two lattice $V_{A_1}$ and $V_{E_8}$, respectively. For $m=3/2$, the allowed central charges are $c=16$ and $c=23\frac{1}{2}$. In the later case the shorter Moonshine module $V\!B^{\natural}$ [@Ho-dr] is a known example. For $m=2$, the allowed central charges are $c=8$, $16$, $23\frac{3}{35}$, $23\frac{1}{2}$, $24$, $32$, $32\frac{4}{5}$, $33\frac{5}{7}$, $34\frac{4}{7}$, $40$, $1496$ and examples are known for $c=8$, $16$, $23\frac{1}{2}$, $24$ and $32$. For conformal