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On entanglement-assisted quantum codes achieving the entanglement-assisted Griesmer bound

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Abstract

The theory of entanglement-assisted quantum error-correcting codes (EAQECCs) is a generalization of the standard stabilizer formalism. Any quaternary (or binary) linear code can be used to construct EAQECCs under the entanglement-assisted (EA) formalism. We derive an EA-Griesmer bound for linear EAQECCs, which is a quantum analog of the Griesmer bound for classical codes. This EA-Griesmer bound is tighter than known bounds for EAQECCs in the literature. For a given quaternary linear code \(\mathcal {C}\), we show that the parameters of the EAQECC that EA-stabilized by the dual of \(\mathcal {C}\) can be determined by a zero radical quaternary code induced from \(\mathcal {C}\), and a necessary condition under which a linear EAQECC may achieve the EA-Griesmer bound is also presented. We construct four families of optimal EAQECCs and then show the necessary condition for existence of EAQECCs is also sufficient for some low-dimensional linear EAQECCs. The four families of optimal EAQECCs are degenerate codes and go beyond earlier constructions. What is more, except four codes, our \([[n,k,d_{ea};c]]\) codes are not equivalent to any \([[n+c,k,d]]\) standard QECCs and have better error-correcting ability than any \([[n+c,k,d]]\) QECCs.

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Acknowledgments

We are indebted to the anonymous reviewers for constructive comments and suggestions on our manuscript, which improve the manuscript significantly. Part of this work was carried out while R. Li was visiting the Chern Institute of Mathematics (CIM) at Nankai University, Tianjin, China. R. Li is grateful to the Institute for the kind hospitality. The research is supported by National Natural Science Foundation of China under Grant No.11471011 and the National Key Basic Research Program of China (“973” program) under Grant No. 2013CB834204.

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Authors and Affiliations

  1. The School of Science, Air Force Engineering University, Xi’an, 710051, Shaanxi, People’s Republic of China

    Ruihu Li & Luobin Guo

  2. Center for Combinatorics, Nankai University, Tianjin, 300071, People’s Republic of China

    Xueliang Li

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  1. Ruihu Li
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  2. Xueliang Li
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Correspondence to Ruihu Li.

Appendices

Appendix A: Proof of Theorem 4.3

Proof

To prove Theorem 4.3, we give our discussion in three steps. Firstly, we construct \(B_{3,i}\) for \(1\le i\le 20\), such that \(G_{5,n}=(G_{5,8}\mid B_{5,i})\) generates a code \(\mathcal {C}_{n}\) and \(R(\mathcal {C}_{n})\) is a two-dimensional code with weight polynomial \(W_{1,n}(z)=1+6z^{4}+9z^{8}\). The matrices \(B_{3,i}\) for \(1\le i\le 20\) are as follows:

$$\begin{aligned} B_{3,1}= & {} \left( \begin{array}{ccccccccccccccccccc} 0\\ 1\\ 3\\ \end{array} \right) , \quad B_{3,2}= \left( \begin{array}{ccccccccccccccccccc} 10\\ 21\\ 02\\ \end{array} \right) , B_{3,3}= \left( \begin{array}{ccccccccccccccccccc} 110\\ 021\\ 103\\ \end{array} \right) , \quad B_{3,4}= \left( \begin{array}{ccccccccccccccccccc} 1 0 0 1\\ 1 1 1 0\\ 0 2 1 2\\ \end{array} \right) ,\\ B_{3,5}= & {} \left( \begin{array}{ccccccccccccccccccc} 11011\\ 31003\\ 12230\\ \end{array} \right) , B_{3,6}=\left( \begin{array}{ccccccccccccccccccc} 1 0 1 1 0 1\\ 2 1 0 0 1 3\\ 0 1 3 1 3 2\\ \end{array} \right) ,\\ B_{3,20}= & {} \left( \begin{array}{ccccccccccccccccccc} 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1\\ 2 3 1 2 0 3 3 0 0 0 2 2 1 1 3 1 0 1 2 0\\ 0 0 2 2 1 3 1 2 3 0 1 1 2 1 2 3 1 0 3 2\\ \end{array} \right) , B_{3,7}=\left( \begin{array}{ccccccccccccccccccc} 0 0 1 1 0 1 1\\ 1 1 2 0 1 0 1\\ 1 3 0 3 2 2 0\\ \end{array} \right) ,\\ B_{3,8}= & {} \left( \begin{array}{ccccccccccccccccccc} 1 1 1 0 0 1 1 1\\ 0 3 2 1 1 0 0 1\\ 2 3 0 2 3 1 3 0\\ \end{array} \right) , \quad B_{3,19}= \left( \begin{array}{ccccccccccccccccccc} 1 0 1 1 1 1 1 0 1 1 1 1 1 0 1 0 1 1 0\\ 0 1 0 2 2 3 3 1 3 2 0 0 3 0 1 1 1 1 0\\ 2 0 1 0 2 0 2 1 1 3 0 3 1 1 0 3 2 3 1\\ \end{array} \right) ,\\ B_{3,9}= & {} \left( \begin{array}{ccccccccccccccccccc} 0 0 1 1 1 1 0 0 1\\ 1 0 1 3 3 1 1 1 3\\ 1 1 3 1 3 0 2 3 0\\ \end{array} \right) , \quad B_{3,10}=\left( \begin{array}{ccccccccccccccccccc} 1 0 0 1 1 1 1 0 1 0\\ 1 1 1 3 3 3 0 1 3 1\\ 0 1 0 0 2 1 2 3 1 2\\ \end{array} \right) ,\\ B_{3,18}= & {} \left( \begin{array}{ccccccccccccccccccc} 1 1 1 1 1 1 1 1 1 0 0 1 1 0 1 1 1 1\\ 3 1 0 0 3 2 3 0 1 1 1 3 2 0 0 1 1 3\\ 3 0 2 1 0 0 2 0 1 3 2 1 1 1 3 3 2 0\\ \end{array} \right) , B_{3,11}=\left( \begin{array}{ccccccccccccccccccc} 0 0 0 1 1 1 1 0 1 1 1\\ 1 1 1 0 2 0 3 0 3 2 3\\ 3 1 2 2 1 1 3 1 0 0 2\\ \end{array} \right) ,\\ B_{3,12}= & {} \left( \begin{array}{ccccccccccccccccccc} 0 0 1 0 1 1 1 1 0 1 1 1\\ 1 1 3 0 2 3 3 2 1 0 0 1\\ 1 2 3 1 1 1 2 0 3 2 1 2\\ \end{array} \right) , \quad B_{3,13}=\left( \begin{array}{ccccccccccccccccccc} 1 1 1 1 1 1 0 1 1 0 1 1 0\\ 2 1 3 0 2 0 1 1 2 1 0 0 1\\ 0 0 2 1 2 2 1 3 1 3 1 3 0\\ \end{array} \right) ,\\ B_{3,14}= & {} \left( \begin{array}{ccccccccccccccccccc} 1 1 1 1 1 1 1 1 1 1 1 1 1 1\\ 0 3 0 3 2 3 3 2 3 1 1 1 2 2\\ 3 2 2 0 0 0 1 1 3 2 1 0 1 2\\ \end{array} \right) , \quad B_{3,17}=\left( \begin{array}{ccccccccccccccccccc} 0 1 0 1 1 1 1 0 1 1 1 0 1 1 0 1 1\\ 1 0 1 0 1 2 0 1 1 3 0 0 2 3 1 2 0\\ 2 3 3 3 0 3 1 1 2 0 2 1 0 1 0 1 1\\ \end{array} \right) ,\\ B_{3,15}= & {} \left( \begin{array}{ccccccccccccccccccc} 0 1 1 1 1 1 0 0 1 1 1 0 1 1 1\\ 1 2 1 0 3 0 1 0 3 3 2 1 2 1 1\\ 3 3 0 3 2 2 1 1 3 1 0 2 0 0 2\\ \end{array} \right) , \quad B_{3,16}=\left( \begin{array}{ccccccccccccccccccc} 1 1 1 1 0 1 1 1 0 1 1 1 0 0 0 1\\ 0 1 0 0 1 3 3 3 1 3 3 0 1 1 1 2\\ 3 2 1 0 2 0 2 1 1 0 2 1 1 3 3 0\\ \end{array} \right) . \end{aligned}$$

Secondly, using the matrices \(B_{3,i}\) for \(1\le i\le 20\), we construct \(G_{5,n}\) for \(n\ge 9\) as follows:

  1. (1)

    If \(n=21t+i\ge 9\) and \(0\le i\le 7\), construct \(G_{5,n}=(G_{5,8}\mid B_{5,13+i}\mid (t-1)D_{5,21})\).

  2. (2)

    \(n=21t+i\ge 9\) and \(8\le i\le 20\), construct \(G_{5,n}=(G_{5,8}\mid B_{5,i-8}\mid tD_{5,21})\).

Let \(\mathcal {C}_{n}\) be the code generated by \(G_{5,n}\), the weight polynomial of \(\mathcal {C}_{n}\) be \(W_{n}(z)\) for \(n\ge 9\). Then \(R(\mathcal {C}_{n})\) is a two-dimensional code with weight polynomial \(W_{R}(z)=1+6z^{4}+9z^{8}\), and all the weight polynomials of these \(W_{n}(z)\) can be derived from \(W_{j}(z)\) for \(8\le j\le 28\). It is not difficult to check \(W_{j}(z)\)’ for \(8\le j\le 28\) are as follows:

$$\begin{aligned} W_{8}(z)= & {} 1+24z^{3}+114z^{4}+144z^{5}+408z^{6}+216z^{7}+117z^{8},\\ W_{9}(z)= & {} 1+54z^{4}+132z^{5}+192z^{6}+360z^{7}+201z^{8}+84z^{9},\\ W_{10}(z)= & {} 1+6z^{4}+96z^{5}+108z^{6}+288z^{7}+273z^{8}+192z^{9}+60z^{10},\\ W_{11}(z)= & {} 1+6z^{4}+144z^{6}+180z^{7}+153z^{8}+360z^{9}+144z^{10}+36z^{11},\\ W_{12}(z)= & {} 1+6z^{4}+12z^{6}+204z^{7}+117z^{8}+264z^{9}+276z^{10}+108z^{11}+36z^{12},\\ W_{13}(z)= & {} 1+6z^{4}+48z^{7}+153z^{8}+276z^{9}+192z^{10}+216z^{11}+96z^{12}+36z^{13},\\ W_{14}(z)= & {} 1+6z^{4}+93z^{8}+216z^{9}+144z^{10}+288z^{11}+180z^{12}+72z^{13}+24z^{14},\\ W_{15}(z)= & {} 1+6z^{4}+9z^{8}+144z^{9}+168z^{10}+252z^{11}+192z^{12}+168z^{13}\\&+\,72z^{14}+12z^{15},\\ W_{16}(z)= & {} 1+6z^{4}+9z^{8}+12z^{9}+216z^{10}+144z^{11}+144z^{12}+348z^{13}\\&+\,72z^{14}+72z^{15},\\ W_{17}(z)= & {} 1+6z^{4}+9z^{8}+72z^{10}+\cdots +12z^{17},\\ W_{18}(z)= & {} 1+6z^{4}+9z^{8}+132z^{11}+\cdots +12z^{18},\\ W_{19}(z)= & {} 1+6z^{4}+9z^{8}+84z^{12}+\cdots +24z^{18},\\ W_{20}(z)= & {} 1+6z^{4}+9z^{8}+168z^{13}+216z^{14}+\cdots +12z^{19},\\ W_{21}(z)= & {} 1+6z^{4}+9z^{8}+84z^{13}+108z^{14}+\cdots +12z^{21},\\ W_{22}(z)= & {} 1+6z^{4}+9z^{8}+120z^{14}+192z^{15}+\cdots +24z^{21},\\ W_{23}(z)= & {} 1+6z^{4}+9z^{8}+156z^{15}+156z^{16}+\cdots +24z^{22},\\ W_{24}(z)= & {} 1+6z^{4}+9z^{8}+48z^{15}+192z^{16}+\cdots +24z^{23},\\ W_{25}(z)= & {} 1+6z^{4}+9z^{8}+48z^{16}+180z^{17}+\cdots +48z^{23},\\ W_{26}(z)= & {} 1+6z^{4}+9z^{8}+60z^{17}+144z^{18}+\cdots +24z^{23},\\ W_{27}(z)= & {} 1+6z^{4}+9z^{8}+144z^{18}+156z^{19}+\cdots +156z^{23},\\ W_{28}(z)= & {} 1+6z^{4}+9z^{8}+192z^{19}+168z^{20}+\cdots +12z^{26} \end{aligned}$$

For \(n=21t+i>28\), from the construction of \(G_{5,n}\), we can deduce weight polynomial \(W_{n}(z)\) of \(\mathcal {C}_{n}\) must be: \(W_{21t+i}(z)=1+6z^{4}+9z^{8}+(W_{21+i}(z)-W_{R}(z))z^{16(t-1)}\) for \(0\le i\le 7\), and \(W_{21t+i}(z)=1+6z^{4}+9z^{8}+(W_{i}(z)-W_{R}(z))z^{16t}\) for \(8\le i\le 20\).

Thirdly, from \(W_{n}(z)\) of \(\mathcal {C}_{n}\) (\(n\ge 9\)), one can deduce the minimal weight \(d_{ea}(n)\)’ of \(\mathcal {C}_{n}\setminus R(\mathcal {C}_{n})\) for \(n=21t+i\), where \(d_{ea}(n)\)’ are as follows:

$$\begin{aligned} d_{ea}(21t+i)= & {} 16t+i-3\quad \hbox {for} \quad t\ge 1, 0\le i\le 2,\\ d_{ea}(21t+i)= & {} 16t+i-4\quad \hbox {for} \quad t\ge 1, 3\le i\le 7,\\ d_{ea}(21t+i)= & {} 16t+i-5\quad \hbox {for} \quad t\ge 0, 8\le i\le 11,\\ d_{ea}(21t+i)= & {} 16t+i-6\quad \hbox {for} \quad t\ge 0, 12\le i\le 15,\\ d_{ea}(21t+i)= & {} 16t+i-7\quad \hbox {for} \quad t\ge 0, 16\le i\le 20. \end{aligned}$$

It is trivial to verify the EAQECCs \([[21t+5,3,16t+1;21t-2]], [[21t+6,3,16t+2;21t-1]]\) and \([[21t+7,3,16t+3;21t]]\) for \(t\ge 1, [[21t+10,3,16t+5;21t+3]], [[21t+11,3,16t+6;21t+4]], [[21t+15,3,16t+9;21t+8]]\) and \([[21t+20,3,16t+13;21t+13]]\) for \(t\ge 0\) achieve the EA-Griesmer bound. The EAQECCs \([[21t,3,16t-3;21t-7]], [[21t+1,3,16t-2;21t-6]], [[21t+2,3,16t-1;21t-5]]\) and \([[21t+8,3,16t+3;21t+1]]\) for \(t\ge 1, [[21t+9,3,16t+4;21t+2]], [[21t+12,3,16t+6;21t+5]], [[21t+13,3,16t+7;21t+6]], [[21t+14,3,16t+8;21t+7]], [[21t+16,3,16t+9;21t+9]], [[21t+17,3,16t+10;21t+10]], [[21t+18,3,16t+11;21t+11]]\) and \([[21t+19,3,16t+12;21t+12]]\) for \(t\ge 0\) are optimal codes and have lengths one above the EA-Griesmer bound. The EAQECCs \([[21t+3,3,16t-1;21t-4]]\) and \([[21t+4,3,16t;21t-3]]\) are optimal codes and have lengths two above the EA-Griesmer bound.

Summarizing the above discussions, the theorem follows.

Appendix B: Proof of Theorem 4.4

Proof

To prove Theorem 4.4, we give our discussion in three steps. Firstly, we construct \(D_{3,i}\) for \(1\le i\le 21\), such that \(G_{6,n}=(G_{6,12}\mid D_{6,i})\) generates a code \(\mathcal {C}_{n}\) and \(R(\mathcal {C}_{n})\) is a three-dimensional code with weight polynomial \(W_{R}(z)=1+9z^{4}+27z^{8}+27z^{12}\). Let \(D_{3,21}=S_{3}\) and construct the matrices \(D_{3,i}\) for \(1\le i\le 20\) as follows:

$$\begin{aligned} D_{3,1}= & {} \left( \begin{array}{ccccccccccccccccccc} 1\\ 3\\ 1\\ \end{array} \right) , \quad D_{3,2}= \left( \begin{array}{ccccccccccccccccccc} 10\\ 01\\ 33\\ \end{array} \right) , \quad D_{3,3}= \left( \begin{array}{ccccccccccccccccccc} 011\\ 011\\ 131\\ \end{array} \right) , \quad D_{3,4}= \left( \begin{array}{ccccccccccccccccccc} 1110\\ 1311\\ 3123\\ \end{array} \right) , \\ D_{3,5}= & {} \left( \begin{array}{ccccccccccccccccccc} 11111\\ 13213\\ 22010 \\ \end{array} \right) , \quad D_{3,9}= \left( \begin{array}{ccccccccccccccccccc} 101011111\\ 011012321\\ 330120223\\ \end{array} \right) ,\\D_{3,20}= & {} \left( \begin{array}{ccccccccccccccccccc} 11111001100111111111\\ 02310100211211133021\\ 33311012223200210302\\ \end{array} \right) , D_{3,6}= \left( \begin{array}{ccccccccccccccccccc} 011101\\ 030113\\ 113322\\ \end{array} \right) ,\\ D_{3,8}= & {} \left( \begin{array}{ccccccccccccccccccc} 10111101\\ 11132001\\ 33112312\\ \end{array} \right) , D_{3,19}= \left( \begin{array}{ccccccccccccccccccc} 1101100111111111111\\ 1311110212311000322 \\ 1103231310302023221\\ \end{array} \right) ,\\ D_{3,7}= & {} \left( \begin{array}{ccccccccccccccccccc} 1001011\\ 1102132\\ 1012223\\ \end{array} \right) , \quad D_{3,10}=\left( \begin{array}{ccccccccccccccccccc} 1110001101\\ 1101111102\\ 1212102010\\ \end{array} \right) ,\\ D_{3,18}= & {} \left( \begin{array}{ccccccccccccccccccc} 001111101101011011\\ 113112111210120120\\ 322332112113321002\\ \end{array} \right) , D_{3,11}=\left( \begin{array}{ccccccccccccccccccc} 10101101111\\ 11011303122\\ 23313213002\\ \end{array} \right) ,\\ D_{3,12}= & {} \left( \begin{array}{ccccccccccccccccccc} 101101101011\\ 113102203101\\ 331213012231\\ \end{array} \right) , \quad D_{3,13}=\left( \begin{array}{ccccccccccccccccccc} 1011110111011\\ 3022321000121\\ 3113101102322\\ \end{array} \right) ,\\ D_{3,14}= & {} \left( \begin{array}{ccccccccccccccccccc} 11111111111111\\ 32313131032020\\ 00311223023213\\ \end{array} \right) , \quad D_{3,17}=\left( \begin{array}{ccccccccccccccccccc} 10101111101111100\\ 31302010111133011\\ 11112203102320032\\ \end{array} \right) ,\\ D_{3,15}= & {} \left( \begin{array}{ccccccccccccccccccc} 101111011101101\\ 111110111013012\\ 133022203001332\\ \end{array} \right) , \quad D_{3,16}=\left( \begin{array}{ccccccccccccccccccc} 1111111111110110\\ 3031112133100021\\ 2000312211331103\\ \end{array} \right) . \end{aligned}$$

Secondly, using the matrices \(D_{3,i}\) for \(1\le i\le 21\), we construct \(G_{6,n}\) for \(n\ge 12\) as follows.

  1. (1)

    If \(n=21t+i\ge 12\) and \(0\le i\le 11\), construct \(G_{6,n}=(G_{6,12}\mid D_{6,9+i}\mid (t-1)D_{6,21})\).

  2. (2)

    \(n=21t+i\ge 12\) and \(12\le i\le 20\), construct \(G_{6,n}=(G_{6,12}\mid D_{6,i-12}\mid tD_{6,21})\).

Let \(\mathcal {C}_{n}\) be the code generated by \(G_{6,n}\), the weight polynomial of \(\mathcal {C}_{n}\) be \(W_{6,n}(z)\) for \(n\ge 12\). Then \(R(\mathcal {C}_{n})\) is a three-dimensional code with weight polynomial \(W_{R}(z)=1+9z^{4}+27z^{8}+27z^{12}\), and all the weight polynomials of these \(W_{n}(z)\) can be derived from \(W_{6,j}(z)\) for \(12\le j\le 32\). It is not difficult to check \(W_{6,j}(z)\)’s for \(12\le j\le 32\) are as follows:

$$\begin{aligned} W_{6,12}(z)= & {} 1+9z^{4}+288z^{6}+432z^{7}+459z^{8}+\cdots +171z^{12},\\ W_{6,13}(z)= & {} 1+9z^{4}+24z^{6}+408z^{7}+387z^{8}+\cdots +120z^{13},\\ W_{6,14}(z)= & {} 1+9z^{4}+96z^{7}+363z^{8}+768z^{9}+\cdots +48z^{14},\\ W_{6,15}(z)= & {} 1+9z^{4}+147z^{8}+648z^{9}+504z^{10} +840z^{11}+\cdots +24z^{15},\\ W_{6,16}(z)= & {} 1+9z^{4}+27z^{8}+288z^{9}+384z^{10} +\cdots +48z^{16},\\ W_{6,17}(z)= & {} 1+9z^{4}+27z^{8}+96z^{9}+240z^{10} +672z^{11}+\cdots ++48z^{17},\\ W_{6,18}(z)= & {} 1+9z^{4}+27z^{8}+96z^{10} +384z^{11}+651z^{12}+\cdots +48z^{18},\\ W_{6,19}(z)= & {} 1+9z^{4}+27z^{8} +288z^{11}+291z^{12}+\cdots +24z^{19},\\ W_{6,20}(z)= & {} 1+9z^{4}+27z^{8}+171z^{12}+816z^{13}+\cdots +576z^{18},\\ W_{6,21}(z)= & {} 1+9z^{4}+27z^{8}+27z^{12}+360z^{13}+672z^{14}+\cdots +24z^{21},\\ W_{6,22}(z)= & {} 1+9z^{4}+27z^{8} +27z^{12}+240z^{13}+384z^{14}+\cdots +96z^{21},\\ W_{6,23}(z)= & {} 1+9z^{4}+27z^{8} +27z^{12}+168z^{14}+408z^{15}+\cdots +24z^{23},\\ W_{6,24}(z)= & {} 1+9z^{4}+27z^{8} +27z^{12}+264z^{15}+\cdots +24z^{23},\\ W_{6,25}(z)= & {} 1+9z^{4}+27z^{8}+27z^{12}+144z^{15}+264z^{16}+\cdots +24z^{23},\\ W_{6,26}(z)= & {} 1+9z^{4}+27z^{8} +27z^{12}+168z^{16}+360z^{17}+\cdots +48z^{25},\\ W_{6,27}(z)= & {} 1+9z^{4}+27z^{8} +27z^{12}+384z^{17}+504z^{18}+\cdots +120z^{25},\\ W_{6,28}(z)= & {} 1+9z^{4}+27z^{8} +27z^{12}+288z^{18}+624z^{19}+\cdots +288z^{25},\\ W_{6,29}(z)= & {} 1+9z^{4}+27z^{8} +27z^{12}+408z^{19}+408z^{20} +\cdots +120z^{26},\\ W_{6,30}(z)= & {} 1+9z^{4}+27z^{8} +27z^{12}+288z^{19}+168z^{20}+\cdots +48z^{28},\\ W_{6,31}(z)= & {} 1+9z^{4}+27z^{8} +27z^{12}+144z^{20}+\cdots +48z^{28},\\ W_{6,32}(z)= & {} 1+9z^{4}+27z^{8} +27z^{12} +480z^{21}+240z^{22}+\cdots +48z^{30}. \end{aligned}$$

Thirdly, from the weight polynomial \(W_{6, n}(z)\) of \(\mathcal {C}_{n}\) with \( n\ge 12\), one can deduce the minimal weight \(d_{ea}(n)\)’ of \(\mathcal {C}_{n}\setminus R(\mathcal {C}_{n})\) for \(n=21t+i\) is as follows:

$$\begin{aligned} d_{ea}(21t+i)= & {} 16t+i-4\quad \hbox {for}\quad t\ge 1, 1\le i\le 3,\\ d_{ea}(21t+i)= & {} 16t+i-5\quad \hbox {for}\quad t\ge 1, 4\le i\le 8,\\ d_{ea}(21t+i)= & {} 16t+i-6\quad \hbox {for}\quad t\ge 1, 9\le i\le 11,\\ d_{ea}(21t+12)= & {} 16t+6\quad \hbox {for}\quad t\ge 0,\\ d_{ea}(21t+i)= & {} 16t+i-7\quad \hbox {for}\quad t\ge 0, 13\le i\le 16,\\ d_{ea}(21t+i)= & {} 16t+i-8\quad \hbox {for}\quad t\ge 0, 17\le i\le 21. \end{aligned}$$

It is trivial to verify the EAQECCs \([[21t,3,16(t-1)+13;21t-3]], [[21t+6,3,16t+1;21t-3]], [[21t+7,3,16t+3;21t-2]], [[21t+8,3,16t+3;21t-1]]\) and \([[21t+11,3,16t+5;21t+2]]\) for \(t\ge 1, [[21t+12,3,16t+6;21t+3]]\) and \([[21t+16,3,16t+9;21t+7]]\) for \(t\ge 0\) achieve the EA-Griesmer bound. The EAQECCs \([[21t+1,3,16t-3;21t-8]], [[21t+2,3,16t-2;21t-7]], [[21t+3,3,16t-1;21t-6]]\), \([[21t+9,3,16t+3;21t]]\) and \([[21t+10,3,16t+4;21t+1]]\) for \(t\ge 1\), and \([[21t+13,3,16t+6;21t+4]], [[21t+14,3,16t+7;21t+5]], [[21t+15,3,16t+8;21t+6]], [[21t+17,3,16t+9;21t+8]], [[21t+18,3,16t+10;21t+9]], [[21t+19,3,16t+11;21t+10]]\) and \([[21t+20,3,16t+12;21t+11]]\) for \(t\ge 0\) are optimal codes and have lengths one above the EA-Griesmer bound. The EAQECCs \([[21t+4,3,16t-1;21t-5]]\) and \([[21t+5,3,16t;21t-4]]\) are optimal codes and have lengths two above the EA-Griesmer bound. Summarizing the above discussions, the theorem follows.

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Li, R., Li, X. & Guo, L. On entanglement-assisted quantum codes achieving the entanglement-assisted Griesmer bound. Quantum Inf Process 14, 4427–4447 (2015). https://doi.org/10.1007/s11128-015-1143-5

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