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Quantum codes from algebraic curves with automorphisms

Tony Shaska

Published Jan 1, 2008• 2 citations
Publisher

Abstract

Let X be an algebraic curve of genus g ≥ 2 defined over a field Fq of characteristic p > 0. From X, under certain conditions, we can construct an algebraic geometry code C. If the code C is self-orthogonal under the symplectic product then we can construct a quantum code Q, called a QAG-code. In this paper we study the construction of such codes from curves with automorphisms and the relation between the automorphism group of the curve X and the codes C and Q.

DOI
10.5488/cmp.11.2.383

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Let X be an algebraic curve of genus g ≥ 2 defined over a field Fq of characteristic p > 0. From X, under certain conditions, we can construct an algebraic geometry code C. If the code C is self-orthogonal under the symplectic product then we can construct a quantum code Q, called a QAG-code. In this paper we study the construction of such codes from curves with automorphisms and the relation between the automorphism group of the curve X and the codes C and Q.
In the present paper, we show that if the dimension of an arbitrary algebraic geometry code over a finite field of even characteristic is slightly less than n/2-g with n being the length of the code and g being the genus of the base curve, then it is equivalent to an Euclidean self-orthogonal code. Previously, such results required a strong condition on the existence of a certain differential. We also show a similar result on Hermitian self-orthogonal algebraic geometry codes. As a consequence, we can apply our result to quantum codes and obtain some good quantum codes. In particular, we obtain a q-ary quantum [[q+1,1]]-MDS code for an even power q which is essential for quantum secret sharing.
This Diplom thesis provides an explicit construction of a quantum Goppa code for any hyperelliptic curve over a non-binary field. Hyperelliptic curves have conjugate pairs of rational places. We use these pairs to construct self-orthogonal classical Goppa codes with respect to a weighted inner product. These codes are also self-orthogonal with respect to a symplectic inner product and therefore define quantum stabilizer codes. A final transformation leads to a quantum Goppa code with respect to the standard symplectic inner product. Some examples illustrate the described construction. Furthermore we present a projection of a higher dimensional code onto the base field and a special case when the projected code is again weighted self-orthogonal and symmetric.
There has been a lot of effort to construct good quantum codes from the classical error correcting codes. Constructing new quantum codes, using Hermitian self-orthogonal codes, seems to be a difficult problem in general. In this paper, Hermitian self-orthogonal codes are studied from algebraic function fields. Sufficient conditions for the Hermitian self-orthogonality of an algebraic geometry code are presented. New Hermitian self-orthogonal codes are constructed from projective lines, elliptic curves, hyper-elliptic curves, Hermitian curves, and Artin-Schreier curves. In addition, over the projective lines, we construct new families of MDS quantum codes with parameters [[N,N−2K,K+1]]q[[N,N-2K,K+1]]_q[[N,N−2K,K+1]]q​ under the following conditions: i) N=t(q−1)+1N=t(q-1)+1N=t(q−1)+1 or t(q−1)+2t(q-1)+2t(q−1)+2 with t∣(q+1)t|(q+1)t∣(q+1) and K=⌊t(q−1)+12t⌋+1K=\lfloor\frac{t(q-1)+1}{2t}\rfloor+1K=⌊2tt(q−1)+1​⌋+1; ii) (n−1)∣(q2−1)(n-1)|(q^2-1)(n−1)∣(q2−1), N=nN=nN=n or N=n+1N=n+1N=n+1, K0=⌊n+q−1q+1⌋K_0=\lfloor\frac{n+q-1}{q+1}\rfloorK0​=⌊q+1n+q−1​⌋, and K≥K0+1K\ge K_0+1K≥K0​+1; iii) N=tq+1N=tq+1N=tq+1, ∀ 1≤t≤q\forall~1\le t\le q∀ 1≤t≤q and K=⌊tq+q−1q+1⌋+1K=\lfloor\frac{tq+q-1}{q+1}\rfloor+1K=⌊q+1tq+q−1​⌋+1; iv) n∣(q2−1)n|(q^2-1)n∣(q2−1), n2=ngcd⁡(n,q+1)n_2=\frac{n}{\gcd (n,q+1)}n2​=gcd(n,q+1)n​, ∀ 1≤t≤q−1n2−1\forall~ 1\le t\le \frac{q-1}{n_2}-1∀ 1≤t≤n2​q−1​−1, N=(t+1)n+2N=(t+1)n+2N=(t+1)n+2 and K=⌊(t+1)n+1+q−1q+1⌋+1K=\lfloor \frac{(t+1)n+1+q-1}{q+1}\rfloor+1K=⌊q+1(t+1)n+1+q−1​⌋+1.
The goal of this paper is twofold. First, to provide a self contained, detailed and rigorous mathematical introduction to some aspects of the quantum error-correcting codes and especially quantum stabilizer codes and their connection to self-orthogonal linear codes. This has been done without venturing that much, if at all, into the world of physics. While most of the results presented are not new, it is not easy to extract a precise mathematical formulation of results and to provide their rigorous proofs by reading the vast number of papers in the field, quite a few of which are written by computer scientists or physicists. It is this formulation and proofs, some of which are new, that we present here. Techniques from algebra of finite fields as well as representations of finite abelian groups have been employed. The second goal is the construction of some stabilizer codes via self-orthogonal linear codes associated to algebraic curves.