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

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.

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) 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) $n=21t+i\ge 9$ 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) 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) $n=21t+i\ge 12$ 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.