Binary codes and vertex operator (super)algebras
The author studies vertex operator algebras whose Virasoro element is a sum of pairwise orthogonal rational conformal vectors with central charge \({1\over2}\), the most important example being the moonshine module \(V^\natural\). In the paper under review the author constructs a simple vertex operator algebra \(M_D\) for an even binary code \(D\subset \mathbb{Z}^n_2\) by using the tensor product of \(n\) copies of the Neveu-Schwarz vertex operator superalgebra \(M=L({1\over2},0)\oplus L({1\over2},{1\over2})\). It is proved that if \(D\) has no words of weight 2, then \((M_D)_1=0\), i.e. there are no nonzero vectors of weight 1, and the full automorphism group of \(M_D\) is finite. Moreover, if \(D\) contains an \([8,4,4]\) Hamming subcode \(C\), then \(C\) defines a nontrivial involutive automorphism of \(M_D\) which is not induced from the automorphism group of \(D\). Some of the proofs rely on results in [\textit{M. Miyamoto}, Griess algebras and conformal vectors in vertex operator algebras, J. Algebra 179, No. 2, 523-548 (1996; Zbl 0964.17021)].
scientific article; zbMATH DE number 883756