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International Symposium on Turbo Codes and Iterative Information Processing

On Propagation Rules for Entanglement-Assisted Quantum Codes

Gaojun Luo, Lin Sok, Martianus Frederic Ezerman, San Ling

Published Sep 4, 2023• 6 citations
Publisher

Abstract

This paper proposes new propagation rules on quantum codes in the entanglement-assisted scenario. The rules lead to new families of such quantum codes whose parameters are demonstrably optimal. To obtain the results, we devise tools to puncture and shorten codes in ways that ensure their Hermitian hulls have certain desirable properties.
DOI
10.1109/istc57237.2023.10273555

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We prove that any Hermitian self-orthogonal [n,k,d]q2[n,k,d]_{q^2}[n,k,d]q2​ code gives rise to an [n,k,d]q2[n,k,d]_{q^2}[n,k,d]q2​ code with ℓ\ellℓ dimensional Hermitian hull for 0≤ℓ≤k0\le \ell \le k0≤ℓ≤k. We present a new method to construct Hermitian self-orthogonal [n,k]q2[n,k]_{q^2}[n,k]q2​ codes with large dimensions k>n+q−1q+1k>\frac{n+q-1}{q+1}k>q+1n+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]q2[n,k]_{q^2}[n,k]q2​ linear codes with Hermitian hull having large dimensions k>n+q−1q+1k>\frac{n+q-1}{q+1}k>q+1n+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.
We craft a special class of unitary matrices overFq2\mathbb{F}q^{2}Fq2to generate classical linear codes whose Hermitian hulls have dimensions that we can design. We then use the codes as classical ingredients in the construction of good entanglement-assisted quantum codes. Over finite fields of characteristic 2, we propose two explicit constructions. To highlight their efficacy, we list excellent qubit codes whose parameters are either new or strictly better than comparable best-known codes in the literature.
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