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arXiv:2110.00769

New families of quantum stabilizer codes from Hermitian self-orthogonal algebraic geometry codes

Lin Sok

Published Oct 2, 2021• 1 citation

Abstract

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 $[[N,N-2K,K+1]]_q$ under the following conditions: i) $N=t(q-1)+1$ or $t(q-1)+2$ with $t|(q+1)$ and $K=\lfloor\frac{t(q-1)+1}{2t}\rfloor+1$; ii) $(n-1)|(q^2-1)$, $N=n$ or $N=n+1$, $K_0=\lfloor\frac{n+q-1}{q+1}\rfloor$, and $K\ge K_0+1$; iii) $N=tq+1$, $\forall~1\le t\le q$ and $K=\lfloor\frac{tq+q-1}{q+1}\rfloor+1$; iv) $n|(q^2-1)$, $n_2=\frac{n}{\gcd (n,q+1)}$, $\forall~ 1\le t\le \frac{q-1}{n_2}-1$, $N=(t+1)n+2$ and $K=\lfloor \frac{(t+1)n+1+q-1}{q+1}\rfloor+1$.

DOI: 10.48550/arxiv.2110.00769

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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

under the following conditions: i)

N=t(q-1)+1

or

t(q-1)+2

with

t|(q+1)

and

K=\lfloor\frac{t(q-1)+1}{2t}\rfloor+1

; ii)

(n-1)|(q^2-1)

,

N=n

or

N=n+1

,

K_0=\lfloor\frac{n+q-1}{q+1}\rfloor

, and

K\ge K_0+1

; iii)

N=tq+1

,

\forall~1\le t\le q

and

K=\lfloor\frac{tq+q-1}{q+1}\rfloor+1

; iv)

n|(q^2-1)

,

n_2=\frac{n}{\gcd (n,q+1)}

,

\forall~ 1\le t\le \frac{q-1}{n_2}-1

,

N=(t+1)n+2

and

K=\lfloor \frac{(t+1)n+1+q-1}{q+1}\rfloor+1

.

We construct codes from rational function fields and provide sufficient conditions for a rational algebraic geometry code to be Hermitian self-orthogonal. A method to embed such codes yields new families of Hermitian self-orthogonal MDS codes with large dimensions. Using the stabilizer formalism, we construct four new families of quantum MDS codes with parameters

[N, N-2 K, K+1]_{q}

. We give conditions on the length N and the values of K of the dimension N-2K. Parameter comparisons over small fields highlight the novelty of the quantum codes that we obtain.

A curve attaining the Hasse-Weil bound is called a maximal curve. Usually, classical error-correcting codes obtained from a maximal curve have good parameters. However, the quantum stabilizer codes obtained from such classical error-correcting codes via Euclidean or Hermitian self-orthogonality do not always possess good parameters. In this paper, the Hermitian self-orthogonality of algebraic geometry codes obtained from two maximal curves is investigated. It turns out that the stabilizer quantum codes produced from such Hermitian self-orthogonal classical codes have good parameters.

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.

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.

We prove that any Hermitian self-orthogonal

[n,k,d]_{q^2}

code gives rise to an

[n,k,d]_{q^2}

code with

\ell

dimensional Hermitian hull for

0\le \ell \le k

. We present a new method to construct Hermitian self-orthogonal

[n,k]_{q^2}

codes with large dimensions

k>\frac{n+q-1}{q+1}

. New families of Hermitian self-orthogonal codes with good parameters are obtained; more precisely those containing almost MDS codes. By applying a puncturing technique to Hermitian self-orthogonal codes, MDS

[n,k]_{q^2}

linear codes with Hermitian hull having large dimensions

k>\frac{n+q-1}{q+1}

are also derived. New families of MDS, almost MDS and optimal codes with arbitrary Hermitian hull dimensions are explicitly constructed from algebraic curves. As an application, we provide entanglement-assisted quantum error correcting codes with new parameters.

Abstract Let q be a prime power. Let

\lambda >1

λ > 1 be a divisor of

q-1

q - 1 and

\tau >1

τ > 1 be a divisor of

q+1

q + 1 . Under certain conditions we prove that there exists an MDS stabilizer quantum code with length

n=\lambda \tau \sigma

n = λ τ σ for a range of values of

\sigma

σ . This is a flexible construction, which includes previously known families and also new MDS parameters not known before.