Algebraic geometric construction of a quantum stabilizer code

Abstract

The stabilizer code is the most general algebraic construction of quantum error-correcting codes proposed so far. A stabilizer code can be constructed from a self-orthogonal subspace of a symplectic space over a finite field. We propose a construction method of such a self-orthogonal space using an algebraic curve. By using the proposed method we construct an asymptotically good sequence of binary stabilizer codes. As a byproduct we improve the Ashikhmin-Litsyn-Tsfasman bound of quantum codes. The main results in this paper can be understood without knowledge of quantum mechanics.Comment: LaTeX2e, 12 pages, 1 color figure. A decoding method was added and several typographical errors were corrected in version 2. The description of the decoding problem was completely wrong in version 1. In version 1 and 2, there was a critical miscalculation in the estimation of parameters of codes, and the constructed sequence of codes turned out to be worse than existing ones. The asymptotically best sequence of quantum codes was added in version 3. Section 3.2 appeared in IEEE Transactions on Information Theory, vol. 48, no. 7, pp. 2122-2124, July 200