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# New quantum codes from metacirculant graphs via self-dual additive $\mathbb{F}_4$-codes

• * Corresponding author: Martianus Frederic Ezerman

Nanyang Technological University Grant Number 04INS000047C230GRT01 supports the research carried out by M. F. Ezerman

• We use symplectic self-dual additive codes over $\mathbb{F}_4$ obtained from metacirculant graphs to construct, for the first time, $\left[\kern-0.15em\left[ {\ell, 0, d} \right]\kern-0.15em\right]$ qubit codes with parameters $(\ell,d) \in \{(78, 20), (90, 21), (91, 22), (93,21),(96,22)\}$. Secondary constructions applied to the qubit codes result in many new qubit codes that perform better than the previous best-known.

Mathematics Subject Classification: Primary: 94B25, 81P73; Secondary: 05C75.

 Citation:

• Figure 1.  The Petersen Graph as $\Gamma(2,5,2,\{1,4\}, \{0\})$

Figure 2.  $G_{12}: = \Gamma(2, 6, 5, \{ 3 \},\{ 0, 3, 4, 5 \})$ of the dodecacode $\mathcal{D}$.

Table 1.  New codes from modifying $Q_{78}$

 Code Parameters Propagation rule $Q_{78, 1}$ $\left[\kern-0.15em\left[ {77,0,19} \right]\kern-0.15em\right]_2$ Puncture $Q_{78}$ at $\{78\}$ $Q_{78,2}$ $\left[\kern-0.15em\left[ {77,1,19} \right]\kern-0.15em\right]_2$ Shorten $Q_{78}$ at $\{78\}$ $Q_{78,3}$ $\left[\kern-0.15em\left[ {78,1,19} \right]\kern-0.15em\right]_2$ Lengthen $Q_{78,2}$ by $1$ $Q_{78,4}$ $\left[\kern-0.15em\left[ {76,2,18} \right]\kern-0.15em\right]_2$ Shorten $Q_{78}$ at $\{77, 78\}$ $Q_{78,5}$ $\left[\kern-0.15em\left[ {76,1,18} \right]\kern-0.15em\right]_2$ Subcode of $Q_{78,4}$ $Q_{78,6}$ $\left[\kern-0.15em\left[ {77,2,18} \right]\kern-0.15em\right]_2$ Lengthen $Q_{78,4}$ by $1$ $Q_{78,7}$ $\left[\kern-0.15em\left[ {75,3,17} \right]\kern-0.15em\right]_2$ Shorten $Q_{78}$ at $\{76,77,78\}$ $Q_{78,8}$ $\left[\kern-0.15em\left[ {76,3,17} \right]\kern-0.15em\right]_2$ Lengthen $Q_{78,7}$ by $1$ $Q_{78,9}$ $\left[\kern-0.15em\left[ {75, 2, 17} \right]\kern-0.15em\right]_2$ Subcode of $Q_{78,7}$

Table 2.  Properties of the Graphs

 $G$ $d_{\rm min}(G)$ $\nu(G)$ $\gamma(G)$ $|{\rm Aut}(G)|$ $G$ $d_{\rm min}(G)$ $\nu(G)$ $\gamma(G)$ $|{\rm Aut}(G)|$ $G_{12}$ $6$ $5$ $4$ $24$ $G_{78}$ $20$ $41$ $7$ $78$ $G_{27,1}$ $9$ $16$ $6$ $27$ $G_{90}$ $21$ $42$ $7$ $90$ $G_{27,2}$ $9$ $10$ $4$ $27$ $G_{91}$ $22$ $44$ $7$ $546$ $G_{36,1}$ $12$ $13$ $6$ $72$ $G_{93}$ $22$ $28$ $4$ $186$ $G_{36,2}$ $12$ $13$ $4$ $72$ $G_{96}$ $22$ $35$ $6$ $96$
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