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On symplectic hulls of linear codes and related applications

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Abstract

This paper investigates symplectic hulls of linear codes. We use a different view to obtain more structural properties of generator matrices with respect to the symplectic inner product. As an outgrowth, generalized formulas for calculating dimensions of symplectic hulls are derived, which extend some known results in the literature. We then study the symplectic hull-variation problem and prove that a monomially equivalent linear code with a smaller dimensional symplectic hull can always be explicitly derived from a given q-ary symplectic self-dual code with a standard generator matrix for \(q\ge 3\). As an application, we present an improved propagation rule for constructing entanglement-assisted quantum error-correcting codes (EAQECCs) and obtain some new and record-breaking binary EAQECCs.

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Acknowledgements

The authors would like to thank the editor and the anonymous referees for their helpful comments and suggestions that greatly improved the presentation and quality of this paper. This work is supported by the National Natural Science Foundation of China under Grant No. 12171134 and U21A20428.

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Appendix

Appendix

Let \(\omega \) be a primitive element of \(\mathbb {F}_4\). The generator matrix of \(\phi (\mathcal {C})\) discussed in Example 5.8 is

$$\begin{aligned} \left( \begin{array}{c|c} 1 0 0 0 0 0 0 0 0 0 0 0 &{} \omega ^2 0 1 \omega ^2 0 \omega \omega ^2 \omega 0 \omega ^2 1 \omega \omega ^2 \omega ^2 \omega 1 \omega ^2 \omega 1 \omega ^2 \omega \omega 1 \omega ^2 1 0 \omega 1 \omega ^2 \omega ^2 \omega ^2 \omega \omega \\ 0 1 0 0 0 0 0 0 0 0 0 0 &{}\omega \omega \omega 1 0 \omega ^2 \omega 1 \omega ^2 \omega \omega ^2 1 0 \omega 1 \omega \omega 0 0 \omega 0 \omega 0 0 1 1 0 0 0 1 1 1 \omega ^2\\ 0 0 1 0 0 0 0 0 0 0 0 0 &{}1 \omega ^2 1 1 \omega 1 0 \omega 0 \omega \omega ^2 1 0 \omega \omega 0 \omega \omega ^2 \omega ^2 0 \omega ^2 \omega 1 \omega \omega ^2 1 \omega ^2 \omega ^2 \omega 1 0 1 \omega \\ 0 0 0 1 0 0 0 0 0 0 0 0 &{}\omega 0 1 1 \omega 1 1 \omega \omega ^2 \omega \omega \omega ^2 \omega ^2 1 0 \omega \omega ^2 1 1 0 \omega ^2 \omega ^2 0 0 0 \omega ^2 1 1 \omega 0 \omega ^2 \omega 0\\ 0 0 0 0 1 0 0 0 0 0 0 0 &{}0 \omega 0 1 1 \omega 1 1 \omega \omega ^2 \omega \omega \omega ^2 \omega ^2 1 0 \omega \omega ^2 1 1 0 \omega ^2 \omega ^2 0 0 0 \omega ^2 1 1 \omega 0 \omega ^2 \omega \\ 0 0 0 0 0 1 0 0 0 0 0 0 &{}\omega 1 0 0 \omega \omega \omega \omega ^2 0 0 \omega ^2 \omega 0 \omega 1 1 \omega ^2 1 1 \omega \omega 0 1 \omega \omega 0 0 1 0 \omega ^2 0 \omega \omega \\ 0 0 0 0 0 0 1 0 0 0 0 0 &{}\omega \omega ^2 \omega ^2 0 \omega ^2 1 \omega 0 \omega \omega 0 \omega ^2 0 1 0 1 \omega 0 \omega ^2 \omega 1 \omega \omega 0 0 \omega 0 \omega 0 \omega 1 \omega \omega ^2\\ 0 0 0 0 0 0 0 1 0 0 0 0 &{}1 \omega ^2 0 0 \omega ^2 \omega \omega \omega 1 \omega ^2 \omega ^2 \omega \omega \omega 1 1 1 \omega ^2 \omega ^2 \omega ^2 \omega ^2 \omega ^2 1 0 \omega ^2 0 0 \omega ^2 0 1 \omega ^2 1 1\\ 0 0 0 0 0 0 0 0 1 0 0 0 &{}\omega ^2 1 \omega \omega ^2 0 1 1 0 \omega \omega \omega 1 1 1 0 0 \omega \omega ^2 \omega 0 1 1 \omega \omega 1 \omega ^2 \omega 1 0 \omega ^2 \omega 1 \omega ^2\\ 0 0 0 0 0 0 0 0 0 1 0 0 &{}1 \omega \omega 1 0 1 \omega 1 1 \omega ^2 \omega ^2 0 0 \omega ^2 1 1 0 \omega ^2 0 \omega 1 \omega ^2 \omega 0 1 1 1 1 \omega ^2 1 \omega \omega \omega \\ 0 0 0 0 0 0 0 0 0 0 1 0 &{}\omega 0 0 \omega \omega \omega ^2 1 0 0 \omega ^2 \omega ^2 \omega ^2 \omega 1 1 1 \omega \omega ^2 1 \omega ^2 1 1 1 \omega ^2 \omega 1 1 \omega ^2 0 1 \omega ^2 0 \omega ^2\\ 0 0 0 0 0 0 0 0 0 0 0 1 &{}1 \omega ^2 \omega ^2 \omega ^2 1 \omega ^2 0 1 1 1 \omega 1 \omega 0 1 0 1 \omega ^2 0 1 \omega \omega ^2 \omega \omega ^2 0 \omega \omega ^2 \omega 1 1 0 \omega ^2 \omega ^2\\ \hline \omega 0 0 0 0 0 0 0 0 0 0 0 &{} \omega 1 \omega 0 \omega ^2 \omega ^2 0 \omega 1 \omega 0 0 \omega 1 \omega 0 \omega ^2 \omega ^2 \omega \omega ^2 \omega ^2 0 \omega 1 \omega 0 0 \omega 1 \omega \omega \omega 1\\ 0 \omega 0 0 0 0 0 0 0 0 0 0 &{}\omega ^2 \omega 0 1 0 1 0 \omega \omega \omega \omega ^2 \omega \omega ^2 1 \omega ^2 \omega ^2 \omega ^2 1 \omega 1 1 1 1 1 0 \omega \omega 1 1 \omega 1 0 0\\ 0 0 \omega 0 0 0 0 0 0 0 0 0 &{}0 \omega ^2 \omega 0 1 0 1 0 \omega \omega \omega \omega ^2 \omega \omega ^2 1 \omega ^2 \omega ^2 \omega ^2 1 \omega 1 1 1 1 1 0 \omega \omega 1 1 \omega 1 0\\ 0 0 0 \omega 0 0 0 0 0 0 0 0 &{}0 0 \omega ^2 \omega 0 1 0 1 0 \omega \omega \omega \omega ^2 \omega \omega ^2 1 \omega ^2 \omega ^2 \omega ^2 1 \omega 1 1 1 1 1 0 \omega \omega 1 1 \omega 1\\ 0 0 0 0 \omega 0 0 0 0 0 0 0 &{}\omega ^2 0 1 0 \omega \omega \omega \omega 1 \omega ^2 \omega ^2 0 1 0 0 \omega \omega 1 \omega 0 \omega ^2 0 0 \omega 0 1 \omega ^2 1 1 1 \omega \omega ^2 0\\ 0 0 0 0 0 \omega 0 0 0 0 0 0 &{}0 \omega ^2 0 1 0 \omega \omega \omega \omega 1 \omega ^2 \omega ^2 0 1 0 0 \omega \omega 1 \omega 0 \omega ^2 0 0 \omega 0 1 \omega ^2 1 1 1 \omega \omega ^2\\ 0 0 0 0 0 0 \omega 0 0 0 0 0 &{}1 1 0 \omega ^2 \omega 1 1 \omega \omega ^2 \omega ^2 0 1 \omega \omega 1 1 0 \omega ^2 1 1 \omega ^2 \omega 0 \omega \omega ^2 \omega \omega \omega 1 0 0 1 1\\ 0 0 0 0 0 0 0 \omega 0 0 0 0 &{}\omega ^2 1 0 \omega ^2 \omega ^2 0 \omega \omega ^2 \omega 0 \omega \omega \omega 1 0 0 \omega \omega \omega \omega \omega ^2 1 \omega ^2 \omega ^2 \omega ^2 \omega ^2 0 \omega ^2 1 \omega \omega ^2 \omega \omega ^2\\ 0 0 0 0 0 0 0 0 \omega 0 0 0 &{}1 \omega \omega \omega ^2 0 \omega \omega ^2 \omega \omega \omega ^2 1 0 \omega ^2 0 1 1 0 \omega ^2 1 \omega \omega ^2 1 \omega 1 0 \omega ^2 1 \omega ^2 1 0 \omega ^2 \omega ^2 1\\ 0 0 0 0 0 0 0 0 0 \omega 0 0 &{}\omega ^2 1 \omega ^2 1 \omega ^2 \omega 1 1 \omega 1 \omega \omega ^2 \omega ^2 0 \omega 0 \omega \omega \omega \omega 0 1 0 1 0 0 1 0 0 \omega \omega ^2 1 1\\ 0 0 0 0 0 0 0 0 0 0 \omega 0 &{}\omega ^2 \omega ^2 0 0 1 1 1 \omega ^2 1 1 0 0 0 0 \omega \omega ^2 \omega ^2 0 \omega ^2 1 0 \omega 0 \omega ^2 0 0 \omega 0 \omega ^2 \omega ^2 1 1 \omega ^2\\ 0 0 0 0 0 0 0 0 0 0 0 \omega &{}1 \omega 0 \omega ^2 \omega ^2 0 \omega 1 \omega 0 0 \omega 1 \omega 0 \omega ^2 \omega ^2 \omega \omega ^2 \omega ^2 0 \omega 1 \omega 0 0 \omega 1 \omega \omega \omega 1 \omega \\ \end{array}\right) \end{aligned}$$

and the generator matrix of \(\phi (\mathcal {C}')\) discussed in Example 5.9 is

$$\begin{aligned} \left( \begin{array}{c|c|c} 1000000000000 &{} 0 &{} 1 \omega ^2 \omega \omega ^2 0 1 0 \omega \omega 0 \omega 0 1 \omega \omega \omega 0 \omega ^2 \omega ^2 \omega ^2 \omega \omega 1 \omega ^2 0 0 1 \omega ^2 1 1 \omega ^2 \omega ^2 1 \omega 0 \omega 0 1 0 \omega \omega ^2 1 \omega 0 \omega \omega ^2 0 1 \omega ^2\\ 0100000000000 &{} \omega &{} \omega ^2 0 \omega 0 \omega ^2 \omega 1 0 \omega ^2 \omega 0 \omega 0 \omega \omega ^2 0 \omega ^2 \omega ^2 \omega ^2 \omega 0 1 \omega \omega 0 \omega ^2 \omega ^2 \omega ^2 1 1 \omega 1 0 \omega 1 \omega ^2 0 \omega ^2 0 1 \omega ^2 \omega ^2 \omega 0 \omega ^2 0 1 0 \omega ^2\\ 0010000000000 &{} \omega &{} 1 \omega 1 0 0 1 \omega 1 1 \omega ^2 \omega 0 \omega \omega ^2 \omega ^2 1 1 0 \omega ^2 \omega 1 \omega ^2 1 1 1 \omega ^2 0 1 1 1 \omega 0 \omega \omega ^2 1 \omega 1 \omega ^2 \omega 1 0 \omega ^2 0 0 \omega ^2 1 \omega 1 \omega \\ 0001000000000 &{} \omega &{} 0 \omega ^2 1 0 0 \omega ^2 1 0 \omega 1 1 \omega 1 \omega ^2 0 \omega ^2 0 1 \omega ^2 1 \omega ^2 0 \omega 1 \omega \omega 1 1 \omega 0 1 \omega ^2 \omega \omega ^2 0 0 0 \omega ^2 \omega 1 \omega ^2 1 \omega \omega 1 \omega \omega ^2 \omega ^2 0\\ 0000100000000 &{} 0 &{} \omega 0 \omega ^2 1 0 0 \omega ^2 1 0 \omega 1 1 \omega 1 \omega ^2 0 \omega ^2 0 1 \omega ^2 1 \omega ^2 0 \omega 1 \omega \omega 1 1 \omega 0 1 \omega ^2 \omega \omega ^2 0 0 0 \omega ^2 \omega 1 \omega ^2 1 \omega \omega 1 \omega \omega ^2 \omega ^2\\ 0000010000000 &{} \omega &{} \omega ^2 \omega ^2 1 1 1 \omega 0 \omega ^2 0 0 \omega 1 1 1 0 1 1 0 0 0 0 \omega \omega ^2 \omega ^2 1 \omega 1 0 \omega ^2 1 1 \omega \omega 0 1 0 \omega \omega ^2 1 \omega \omega ^2 1 0 \omega ^2 1 0 \omega ^2 \omega 1\\ 0000001000000 &{} 0 &{} \omega ^2 0 1 \omega 1 0 \omega \omega 1 0 \omega \omega 0 \omega ^2 \omega ^2 \omega 1 \omega \omega ^2 \omega ^2 \omega \omega \omega ^2 0 \omega ^2 1 \omega ^2 \omega 1 \omega \omega \omega \omega ^2 0 0 \omega ^2 0 \omega ^2 \omega ^2 \omega ^2 1 \omega \omega ^2 0 1 \omega 0 \omega 1\\ 0000000100000 &{} 0 &{} 1 0 \omega \omega \omega 0 0 0 0 1 \omega \omega \omega ^2 \omega 1 1 \omega \omega 1 0 1 0 \omega ^2 0 0 \omega ^2 0 0 \omega ^2 0 1 1 \omega ^2 1 0 \omega \omega ^2 1 \omega ^2 1 0 0 0 \omega ^2 \omega \omega \omega 1 1\\ 0000000010000 &{} 0 &{} 1 \omega \omega 1 \omega \omega ^2 0 \omega \omega 0 \omega ^2 \omega \omega ^2 1 0 \omega ^2 1 1 1 \omega \omega \omega ^2 1 0 0 0 \omega \omega ^2 1 \omega \omega ^2 \omega 0 1 1 \omega \omega \omega 1 1 \omega 1 \omega 0 1 1 \omega \omega ^2 \omega \\ 0000000001000 &{} \omega &{} \omega \omega ^2 1 \omega ^2 1 1 \omega ^2 \omega 1 \omega \omega \omega ^2 \omega ^2 \omega \omega 0 \omega 1 \omega \omega ^2 \omega ^2 \omega ^2 \omega 1 \omega ^2 \omega ^2 \omega \omega ^2 0 0 \omega \omega 0 1 \omega 0 0 0 \omega ^2 \omega \omega ^2 \omega ^2 0 0 1 0 \omega ^2 \omega ^2 \omega \\ 0000000000100 &{} \omega &{} 0 0 0 0 \omega ^2 \omega 1 1 1 1 0 \omega \omega \omega 1 \omega 1 \omega \omega 0 \omega \omega \omega \omega \omega 0 1 \omega ^2 0 1 0 \omega ^2 0 1 \omega \omega ^2 \omega \omega 1 0 0 \omega \omega \omega 1 0 \omega \omega \omega \\ 0000000000010 &{} \omega &{} 0 \omega \omega ^2 1 0 0 \omega \omega ^2 \omega 1 \omega ^2 0 \omega ^2 \omega ^2 1 1 \omega ^2 1 1 0 1 \omega ^2 \omega ^2 \omega 1 1 \omega 0 0 1 1 1 1 1 \omega \omega ^2 1 0 \omega ^2 \omega \omega 1 \omega ^2 0 \omega ^2 0 \omega \omega ^2 \omega ^2\\ 0000000000001 &{} \omega &{} 1 1 \omega ^2 1 1 \omega 0 \omega \omega \omega 1 \omega ^2 0 0 \omega \omega ^2 0 0 1 0 \omega ^2 \omega \omega ^2 0 1 \omega \omega 0 \omega 0 \omega \omega ^2 \omega \omega \omega 1 1 \omega 1 \omega \omega ^2 \omega \omega 1 \omega ^2 1 \omega \omega 1\\ \hline \omega 000000000000 &{} \omega &{} \omega \omega \omega ^2 1 0 \omega ^2 0 \omega \omega ^2 0 \omega 0 1 1 \omega ^2 0 1 0 \omega ^2 \omega 1 1 1 0 \omega ^2 \omega ^2 \omega 1 \omega ^2 1 0 1 \omega 1 \omega ^2 1 \omega \omega 1 \omega ^2 \omega 1 1 \omega 1 1 \omega 1 1\\ 0\omega 00000000000 &{} 0 &{} \omega ^2 1 0 0 1 1 \omega ^2 \omega 0 \omega ^2 \omega \omega 1 \omega ^2 \omega ^2 1 0 \omega \omega ^2 0 0 \omega ^2 0 \omega 0 \omega ^2 \omega 1 0 \omega \omega \omega ^2 0 0 1 1 1 \omega ^2 \omega \omega ^2 0 \omega ^2 \omega ^2 1 0 \omega 1 \omega ^2 \omega \\ 00\omega 0000000000 &{} \omega &{} \omega 1 \omega 1 0 \omega 1 1 1 0 1 \omega \omega ^2 0 0 \omega ^2 0 0 1 1 1 1 \omega 0 1 \omega ^2 1 \omega ^2 \omega 1 \omega \omega ^2 1 1 \omega ^2 0 \omega ^2 \omega ^2 \omega 1 1 1 \omega 1 0 1 0 0 \omega \\ 000\omega 000000000 &{} \omega &{} 0 0 \omega \omega ^2 1 \omega ^2 \omega \omega ^2 \omega 1 \omega 1 \omega ^2 \omega \omega ^2 0 \omega 0 \omega ^2 \omega ^2 0 0 0 \omega \omega ^2 \omega 1 0 0 \omega ^2 1 \omega ^2 1 0 \omega \omega \omega 1 \omega 1 \omega ^2 0 0 0 0 1 \omega ^2 1 1\\ 0000\omega 00000000 &{} 0 &{} \omega ^2 \omega ^2 \omega 1 \omega ^2 0 \omega ^2 0 1 \omega \omega ^2 \omega 0 1 0 1 0 1 \omega ^2 0 1 \omega 1 \omega ^2 \omega \omega ^2 \omega ^2 \omega 1 1 0 \omega \omega \omega ^2 0 0 \omega \omega ^2 1 0 \omega \omega \omega 0 \omega \omega ^2 1 \omega \omega \\ 00000\omega 0000000 &{} \omega &{} \omega 1 0 \omega ^2 1 0 0 1 \omega ^2 1 0 \omega ^2 \omega ^2 1 \omega 0 0 0 \omega 1 1 0 \omega ^2 1 0 1 1 \omega 1 0 1 1 0 \omega ^2 0 1 \omega 0 \omega \omega \omega \omega ^2 \omega ^2 0 1 \omega ^2 1 0 \omega ^2\\ 000000\omega 000000 &{} \omega &{} 1 \omega ^2 0 \omega \omega ^2 \omega ^2 0 0 0 \omega ^2 1 0 \omega ^2 0 0 0 1 \omega ^2 0 \omega ^2 \omega \omega 0 0 \omega \omega ^2 \omega \omega ^2 0 1 \omega ^2 \omega ^2 \omega \omega ^2 0 \omega ^2 \omega ^2 1 1 \omega ^2 \omega ^2 \omega 0 1 \omega ^2 \omega ^2 1 1 \omega \\ 0000000\omega 00000 &{} \omega &{} 0 \omega ^2 0 1 \omega 0 \omega ^2 \omega \omega ^2 0 1 1 1 \omega \omega ^2 0 1 1 0 \omega \omega \omega ^2 \omega ^2 0 \omega ^2 1 1 \omega ^2 0 1 1 \omega 1 \omega ^2 0 1 1 1 0 \omega 1 \omega \omega ^2 \omega 0 \omega 1 0 0\\ 00000000\omega 0000 &{} 0 &{} \omega 0 \omega ^2 0 1 \omega 0 \omega ^2 \omega \omega ^2 0 1 1 1 \omega \omega ^2 0 1 1 0 \omega \omega \omega ^2 \omega ^2 0 \omega ^2 1 1 \omega ^2 0 1 1 \omega 1 \omega ^2 0 1 1 1 0 \omega 1 \omega \omega ^2 \omega 0 \omega 1 0\\ 000000000\omega 000 &{} 0 &{} 0 \omega 0 \omega ^2 0 1 \omega 0 \omega ^2 \omega \omega ^2 0 1 1 1 \omega \omega ^2 0 1 1 0 \omega \omega \omega ^2 \omega ^2 0 \omega ^2 1 1 \omega ^2 0 1 1 \omega 1 \omega ^2 0 1 1 1 0 \omega 1 \omega \omega ^2 \omega 0 \omega 1\\ 0000000000\omega 00 &{} 0 &{} 1 \omega ^2 0 \omega ^2 \omega ^2 1 1 0 \omega \omega ^2 0 \omega ^2 1 \omega ^2 \omega ^2 \omega ^2 \omega 0 \omega ^2 \omega \omega ^2 \omega \omega ^2 1 \omega ^2 \omega ^2 1 0 0 0 0 \omega ^2 0 \omega ^2 \omega \omega ^2 \omega ^2 1 1 \omega ^2 \omega 1 0 1 0 0 \omega 1 1\\ 00000000000\omega 0 &{} 0 &{} 1 \omega 1 \omega ^2 \omega ^2 \omega 1 \omega ^2 \omega \omega 1 0 \omega \omega ^2 1 1 \omega ^2 1 \omega ^2 0 0 1 \omega ^2 0 1 \omega ^2 \omega \omega 1 1 \omega ^2 \omega ^2 \omega \omega \omega ^2 0 \omega ^2 \omega 1 \omega ^2 0 \omega ^2 \omega ^2 0 \omega ^2 \omega ^2 0 \omega ^2 \omega \\ 000000000000\omega &{} \omega &{} \omega \omega ^2 1 0 \omega ^2 0 \omega \omega ^2 0 \omega 0 1 1 \omega ^2 0 1 0 \omega ^2 \omega 1 1 1 0 \omega ^2 \omega ^2 \omega 1 \omega ^2 1 0 1 \omega 1 \omega ^2 1 \omega \omega 1 \omega ^2 \omega 1 1 \omega 1 1 \omega 1 1 \omega \\ \hline 0000000000000 &{} 1 &{}\omega ^2 \omega \omega ^2 0 1 0 \omega \omega 0 \omega 0 1 \omega \omega \omega 0 \omega ^2 \omega ^2 \omega ^2 \omega \omega 1 \omega ^2 0 0 1 \omega ^2 1 1 \omega ^2 \omega ^2 1 \omega 0 \omega 0 1 0 \omega \omega ^2 1 \omega 0 \omega \omega ^2 0 1 \omega ^2 1\\ \end{array}\right) . \end{aligned}$$

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Li, Y., Zhu, S. On symplectic hulls of linear codes and related applications. J. Appl. Math. Comput. 70, 2603–2622 (2024). https://doi.org/10.1007/s12190-024-02058-8

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