International Symposium on Turbo Codes and Iterative Information Processing

Four New Families of Quantum Stabilizer Codes from Hermitian Self-Orthogonal MDS codes

Lin Sok, Martianus Frederic Ezerman, San Ling

Published Sep 4, 2023• 6 citations

Abstract

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.

DOI: 10.1109/istc57237.2023.10273565

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International Symposium on Turbo Codes and Iterative Information Processing

Four New Families of Quantum Stabilizer Codes from Hermitian Self-Orthogonal MDS codes

Lin Sok, Martianus Frederic Ezerman, San Ling

Published Sep 4, 2023• 6 citations

Publisher

Abstract

[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.

DOI: 10.1109/istc57237.2023.10273565

Related papers

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

Lin Sok

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Lingfei Jin, Chaoping Xing

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.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.

arXiv:1308.3575IEEE Transactions on Information TheoryDec 2, 201178 citations

[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.

[[N,N-2K,K+1]]_q

under the following conditions: i)

N=t(q-1)+1

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

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

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.

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.