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MDS codes with Euclidean and Hermitian hulls of flexible dimensions and their applications to EAQECCs

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Yang Li at Nanyang Technological University
Yang Li
Ruhao Wan
Ruhao Wan
Ruhao Wan
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Shixin Zhu
Shixin Zhu
Shixin Zhu
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Abstract

The hull of a linear code is the intersection of itself with its dual code with respect to certain inner product. Both Euclidean and Hermitian hulls are of theorical and practical significance. In this paper, we construct several new classes of maximum distance separable (MDS) codes via (extended) generalized Reed-Solomon (GRS) codes and determine their Euclidean or Hermitian hulls. As a consequence, four new classes of MDS codes with Hermitian hulls of flexible dimensions and six new classes of MDS codes with Euclidean hulls of flexible dimensions are constructed. As applications, for the former, we further construct four new families of entanglement-assisted quantum error-correcting codes (EAQECCs) and four new families of MDS EAQECCs of length n>q+1n>q+1n>q+1. Meanwhile, many of the distance parameters of our MDS EAQECCs are greater than ⌈q2⌉⌈q2⌉\lceil \frac{q}{2} \rceil or q; for the latter, we show some examples on Euclidean self-orthogonal and one-dimensional Euclidean hull MDS codes. In addition, two new general methods for constructing extended GRS codes with (k-1)(k−1)(k-1)-dimensional Hermitian hull and Hermitian self-orthogonal extended GRS codes are also provided.
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Quantum Information Processing (2023) 22:153
https://doi.org/10.1007/s11128-023-03900-x
MDS codes with Euclidean and Hermitian hulls of flexible
dimensions and their applications to EAQECCs
Yang Li1·Ruhao Wan1·Shixin Zhu1
Received: 13 November 2022 / Accepted: 27 February 2023 / Published online: 22 March 2023
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023
Abstract
The hull of a linear code is the intersection of itself with its dual code with respect to
certain inner product. Both Euclidean and Hermitian hulls are of theorical and practi-
cal significance. In this paper, we construct several new classes of maximum distance
separable (MDS) codes via (extended) generalized Reed-Solomon (GRS) codes and
determine their Euclidean or Hermitian hulls. As a consequence, four new classes of
MDS codes with Hermitian hulls of flexible dimensions and six new classes of MDS
codes with Euclidean hulls of flexible dimensions are constructed. As applications,
for the former, we further construct four new families of entanglement-assisted quan-
tum error-correcting codes (EAQECCs) and four new families of MDS EAQECCs of
length n>q+1. Meanwhile, many of the distance parameters of our MDS EAQECCs
are greater than q
2or q; for the latter, we show some examples on Euclidean self-
orthogonal and one-dimensional Euclidean hull MDS codes. In addition, two new
general methods for constructing extended GRS codes with (k−1)-dimensional
Hermitian hull and Hermitian self-orthogonal extended GRS codes are also provided.
Keywords Hulls ·Entanglement-assisted quantum error-correcting codes ·
Generalized Reed–Solomon codes ·Extended generalized Reed–Solomon codes
Mathematics Subject Classification 94B05 ·81P70
This research was supported by the National Natural Science Foundation of China (Nos.U21A20428 and
12171134).
BShixin Zhu
zhushixinmath@hfut.edu.cn
Yang Li
yanglimath@163.com
Ruhao Wan
wanruhao98@163.com
1School of Mathematics, HeFei University of Technology, Hefei 230601, China
123
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... Naturally, there is a problem: how to construct self-dual codes? In recent year, authors studied self-dual (MDS) codes constructed by GRS codes [7,8,13,14,28,25,15,19] and TGRS codes [9,3,30,22,29]. For a 1-TGRS code to be self-dual, Huang et al. presented a condition that is both sufficient and necessary in [12]. ...
View
... For more details on the encoding procedure, one can see [23,30]. Moreover, an [[n, k, d; c]] q EAQECC is just an [[n, k, d]] q QECC if c = 0. Note that the introduction of EAQECCs have sparked another boom in the research of quantum coding theory and many interesting EAQECCs have been constructed in the literature (see for example [4,5,8,9,12,13,16,17,19,24,26]). ...
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