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Linear codes with arbitrary dimensional hull and their applications to EAQECCs

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Abstract

The Euclidean hull dimension of a linear code is an important quantity to determine the parameters of an entanglement-assisted quantum error-correcting code (EAQECC) if the Euclidean construction is applied. In this paper, we study the Euclidean hull of a linear code by means of orthogonal matrices. We provide some methods to construct linear codes over \({\mathbb {F}}_{p^m}\) with hull of arbitrary dimensions. With existence of self-dual bases of \({\mathbb {F}}_{p^m}\) over \({\mathbb {F}}_p\), we determine a Gray map from \({\mathbb {F}}_{p^m}\) to \({\mathbb {F}}_p^m\), and from a given linear code over \({\mathbb {F}}_{p^m}\) with one-dimensional hull, we construct, using such a Gray map, a linear code over \({\mathbb {F}}_p\) with m-dimensional hull for all m when p is even and for all m odd when p is odd. Comparisons with classical constructions are made, and some good EAQECCs over \({\mathbb {F}}_q, q=2,3,4,5,9,13,17,49\) are presented.

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Acknowledgements

We would like to thank the anonymous referees for their careful reading, constructive comments, and helpful suggestions that have improved the quality of the paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, Anhui University, Hefei, 230601, China

    Lin Sok & Gang Qian

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  1. Lin Sok
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This research work is supported by Anhui Provincial Natural Science Foundation with Grant Number 1908085MA04.

Appendix

Appendix

In the following, we provide some generator matrices of 1-dim hull \([13,4,5]_2\) code, 2-dim hull \([20,10,5]_2\) code, 1-dim hull \([20,15,3]_2\) code, 4-dim hull \([14,10,3]_3\) code, 4-dim hull \([16,12,3]_3\) code, 6-dim hull \([20,14,4]_3\) code, 4-dim hull \([20,16,3]_3\) code, 4-dim hull \([22,18,3]_3\) code, 0-dim hull \([15,9,5]_5\) code, 0-dim hull \([12,8,4]_{13}\) code, 1-dim hull \([18,9,6]_5\) code, and 1-dim hull \([18,9,8]_9\) code. The generator matrices over \({\mathbb {F}}_q\) are expressed in the form \((I|A_q)\), where the identity matrix is omitted.

$$\begin{aligned} \left( \begin{array}{cccccccccc} 0&{}1&{}0&{}0&{}1&{}1&{}1&{}0&{}1\\ 1&{}1&{}0&{}0&{}1&{}0&{}1&{}0&{}0\\ 0&{}1&{}1&{}0&{}0&{}1&{}1&{}0&{}0\\ 0&{}0&{}1&{}1&{}1&{}1&{}0&{}0&{}1\\ \end{array} \right) _2, \left( \begin{array}{cccccccccc} 1&{}1&{}1&{}1&{}1&{}1&{}0&{}1&{}1&{}0\\ 1&{}1&{}1&{}0&{}1&{}0&{}0&{}1&{}0&{}0\\ 0&{}1&{}0&{}1&{}0&{}1&{}0&{}1&{}0&{}1\\ 0&{}0&{}1&{}1&{}1&{}1&{}0&{}0&{}0&{}0\\ 0&{}1&{}1&{}0&{}1&{}0&{}0&{}0&{}1&{}0\\ 0&{}1&{}1&{}1&{}1&{}0&{}1&{}0&{}0&{}1\\ 0&{}1&{}1&{}1&{}0&{}1&{}1&{}1&{}1&{}0\\ 1&{}1&{}1&{}1&{}1&{}1&{}1&{}0&{}1&{}1\\ 0&{}1&{}0&{}0&{}0&{}1&{}0&{}0&{}1&{}1\\ 0&{}0&{}0&{}0&{}0&{}1&{}1&{}1&{}1&{}0\\ \end{array} \right) _2, \left( \begin{array}{cccccccccc} 1&{}0&{}0&{}1&{}0\\ 1&{}1&{}0&{}1&{}0\\ 1&{}0&{}0&{}0&{}1\\ 0&{}1&{}1&{}1&{}1\\ 0&{}1&{}0&{}0&{}1\\ 0&{}0&{}0&{}1&{}1\\ 0&{}1&{}1&{}1&{}0\\ 1&{}1&{}0&{}0&{}1\\ 1&{}1&{}1&{}0&{}0\\ 0&{}1&{}0&{}1&{}0\\ 1&{}1&{}1&{}1&{}0\\ 1&{}0&{}1&{}1&{}1\\ 1&{}0&{}1&{}0&{}0\\ 1&{}1&{}0&{}1&{}1\\ 1&{}0&{}0&{}1&{}1\\ \end{array} \right) _2, \left( \begin{array}{cccccccccc} 2&{}0&{}1&{}1\\ 2&{}2&{}2&{}1\\ 2&{}2&{}0&{}2\\ 0&{}1&{}2&{}1\\ 2&{}0&{}2&{}0\\ 1&{}2&{}1&{}1\\ 1&{}2&{}2&{}0\\ 2&{}0&{}0&{}1\\ 1&{}0&{}2&{}1\\ 0&{}0&{}1&{}1\\ \end{array} \right) _3, \left( \begin{array}{cccccccccc} 0&{}0&{}1&{}1\\ 2&{}2&{}0&{}0\\ 1&{}2&{}2&{}0\\ 1&{}1&{}0&{}1\\ 1&{}2&{}2&{}2\\ 0&{}2&{}1&{}1\\ 2&{}1&{}0&{}2\\ 0&{}0&{}2&{}1\\ 1&{}0&{}1&{}1\\ 2&{}2&{}2&{}0\\ 1&{}0&{}0&{}1\\ 0&{}1&{}1&{}0\\ \end{array} \right) _3, \end{aligned}$$
$$\begin{aligned} \left( \begin{array}{cccccccccc} 1&{}1&{}0&{}0&{}0&{}2\\ 2&{}2&{}2&{}1&{}0&{}2\\ 1&{}2&{}0&{}2&{}2&{}0\\ 1&{}1&{}2&{}2&{}2&{}0\\ 1&{}2&{}1&{}2&{}0&{}0\\ 1&{}2&{}1&{}1&{}1&{}2\\ 0&{}1&{}0&{}2&{}0&{}2\\ 1&{}0&{}1&{}2&{}1&{}2\\ 0&{}2&{}1&{}0&{}0&{}2\\ 1&{}0&{}0&{}1&{}0&{}2\\ 0&{}2&{}2&{}2&{}2&{}2\\ 0&{}1&{}2&{}0&{}1&{}2\\ 0&{}2&{}0&{}2&{}1&{}1\\ 0&{}0&{}0&{}2&{}2&{}2\\ \end{array} \right) _3, \left( \begin{array}{cccccccccc} 1&{}1&{}1&{}2\\ 1&{}2&{}1&{}2\\ 2&{}2&{}0&{}2\\ 2&{}1&{}1&{}1\\ 2&{}0&{}2&{}2\\ 1&{}0&{}0&{}2\\ 2&{}1&{}1&{}2\\ 1&{}2&{}0&{}0\\ 0&{}1&{}1&{}2\\ 2&{}0&{}1&{}1\\ 1&{}2&{}0&{}2\\ 1&{}0&{}2&{}1\\ 1&{}2&{}1&{}1\\ 1&{}0&{}2&{}0\\ 2&{}2&{}2&{}2\\ 0&{}1&{}0&{}2\\ \end{array} \right) _3, \left( \begin{array}{cccccccccc} 0&{}1&{}1&{}0\\ 1&{}2&{}0&{}0\\ 1&{}1&{}2&{}2\\ 0&{}2&{}1&{}2\\ 1&{}0&{}1&{}0\\ 2&{}1&{}2&{}0\\ 1&{}0&{}0&{}2\\ 1&{}1&{}1&{}0\\ 2&{}2&{}2&{}2\\ 1&{}0&{}2&{}1\\ 1&{}1&{}0&{}0\\ 0&{}1&{}0&{}2\\ 1&{}1&{}2&{}0\\ 1&{}0&{}1&{}1\\ 1&{}2&{}1&{}1\\ 0&{}1&{}2&{}2\\ 2&{}1&{}2&{}1\\ 2&{}1&{}1&{}1\\ \end{array} \right) _3, \left( \begin{array}{cccccccccc} 1&{}3&{}2&{}2&{}0&{}1\\ 1&{}1&{}2&{}1&{}4&{}2\\ 3&{}1&{}0&{}1&{}2&{}1\\ 3&{}3&{}1&{}4&{}0&{}2\\ 1&{}1&{}3&{}3&{}1&{}0\\ 3&{}4&{}3&{}1&{}3&{}4\\ 1&{}4&{}1&{}3&{}0&{}2\\ 3&{}0&{}3&{}2&{}1&{}0\\ 0&{}3&{}2&{}0&{}2&{}1\\ \end{array} \right) _5, \left( \begin{array}{cccccccccc} 9&{}7&{}4&{}2\\ 2&{}10&{}7&{}10\\ 11&{}6&{}5&{}11\\ 4&{}5&{}11&{}8\\ 5&{}11&{}12&{}6\\ 6&{}7&{}4&{}5\\ 11&{}5&{}9&{}5\\ 5&{}1&{}3&{}2\\ \end{array} \right) _{13} \end{aligned}$$
$$\begin{aligned} \left( \begin{array}{cccccccccc} 1&{}1&{}1&{}1&{}3&{}3&{}4&{}3&{}2\\ 0&{}4&{}0&{}3&{}1&{}0&{}3&{}0&{}4\\ 3&{}4&{}3&{}1&{}0&{}0&{}4&{}2&{}1\\ 3&{}3&{}4&{}3&{}4&{}4&{}1&{}1&{}3\\ 2&{}2&{}1&{}3&{}2&{}3&{}0&{}3&{}4\\ 2&{}4&{}0&{}4&{}4&{}0&{}3&{}3&{}1\\ 0&{}3&{}4&{}3&{}3&{}2&{}3&{}4&{}3\\ 2&{}3&{}3&{}4&{}4&{}2&{}0&{}3&{}3\\ 0&{}2&{}4&{}2&{}0&{}4&{}3&{}4&{}3\\ \end{array} \right) _{5}, \left( \begin{array}{cccccccccc} 0&{}0&{}w^7&{}w^6&{}2&{}1&{}w^5&{}w^7&{}w\\ 1&{}w^5&{}w^3&{}w^3&{}w^7&{}2&{}2&{}w^7&{}2\\ w^3&{}w^7&{}0&{}0&{}1&{}1&{}1&{}1&{}w^7\\ w^2&{}1&{}2&{}w^7&{}w^2&{}w^6&{}w^2&{}w^5&{}0\\ w^2&{}w^3&{}w^6&{}w^6&{}w^7&{}w^6&{}w^7&{}1&{}2\\ w&{}w^3&{}w^7&{}w^7&{}w^2&{}1&{}w^2&{}w^6&{}w^3\\ w^6&{}0&{}2&{}w^5&{}w^6&{}w^2&{}w^7&{}1&{}w^2\\ 1&{}w^5&{}w^5&{}w^7&{}w^2&{}w^7&{}2&{}w&{}w^3\\ 2&{}w^7&{}w^5&{}0&{}w^5&{}w^3&{}w^3&{}2&{}w\\ \end{array} \right) _{9} \end{aligned}$$

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Sok, L., Qian, G. Linear codes with arbitrary dimensional hull and their applications to EAQECCs. Quantum Inf Process 21, 72 (2022). https://doi.org/10.1007/s11128-021-03407-3

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