Binary Codes and Vertex Operator (Super)Algebras Original Research Article

We study a vertex operator algebra whose Virasoro element is a sum of pairwise orthogonal rational conformal vectors with central charge 1/2. The most important example is the moonshine moduleVmusic natural. In particular, we construct a series of vertex operator algebras whose full automorphism groups are finite. Namely, we construct vertex operator algebrasMD=∑∞i=0(MD)ifrom even linear binary codeDsubset of or equal toimagen2and prove that if the minimum weight ofDis greater than 2 then (MD)1=0 and the full automorphism group ofMDis finite. From the viewpoint of finite group theory, the construction of a vertex operator algebra has one advantage. We can expect not only automorphisms ofD, but also another one. Indeed, ifDcontains a [8, 4, 4] Hamming subcodeC, thenCdefines a nontrivial involutive automorphism ofMD, which is not induced from the automorphism group ofD. In particular, we have a finite group extension of Aut(D)(imagen2/Dperpendicular).