An E8E_8-approach to the moonshine vertex operator algebra

In this article, we study the moonshine vertex operator algebra starting with the tensor product of three copies of the vertex operator algebra $V_{\sqrt2E_8}^+$, and describe it by the quadratic space over \F_2 associated to $V_{\sqrt2E_8}^+$. Using quadratic spaces and orthogonal groups, we show the transitivity of the automorphism group of the moonshine vertex operator algebra on the set of all full vertex operator subalgebras isomorphic to the tensor product of three copies of $V_{\sqrt2E_8}^+$, and determine the stabilizer of such a vertex operator subalgebra. Our approach is a vertex operator algebra analogue of "An $E_8$-approach to the Leech lattice and the Conway group" by Lepowsky and Meurman. Moreover, we find new analogies among the moonshine vertex operator algebra, the Leech lattice and the extended binary Golay code.Comment: 25 page

On the structure of framed vertex operator algebras and their pointwise frame stabilizers

In this paper, we study the structure of a general framed vertex operator algebra. We show that the structure codes (C,D) of a framed VOA V satisfy certain duality conditions. As a consequence, we prove that every framed VOA is a simple current extension of the associated binary code VOA V_C. This result would give a prospect on the classification of framed vertex operator algebras. In addition, the pointwise frame stabilizer of V is studied. We completely determine all automorphisms in this pointwise stabilizer, which are of order 1, 2 or 4. The 4A-twisted sector and the 4A-twisted orbifold theory of the famous Moonshine VOA are also constructed explicitly. We verify that the top module of this twisted sector is of dimension 1 and of weight 3/4 and the VOA obtained by 4A-twisted orbifold construction of the moonshine VOA is isomorphic to the moonshine VOA itself.Comment: Version 3: 59 pages. Corrected version. 54 pages on my LaTeX system version 2: We add Theorem 5.16 in which we give a necessary and sufficient condtion for a code to be a structure code of a holomorphic framed VOA. "hyperref" style is also introduce

On optimal quantum codes

We present families of quantum error-correcting codes which are optimal in the sense that the minimum distance is maximal. These maximum distance separable (MDS) codes are defined over q-dimensional quantum systems, where q is an arbitrary prime power. It is shown that codes with parameters [[n,n-2d+2,d]]_q exist for all 3 <= n <= q and 1 <= d <= n/2+1. We also present quantum MDS codes with parameters [[q^2,q^2-2d+2,d]]_q for 1 <= d <= q which additionally give rise to shortened codes [[q^2-s,q^2-2d+2-s,d]]_q for some s.Comment: Accepted for publication in the International Journal of Quantum Informatio