A small table of upper bounds on the size of EA-CWS codes

Here is a table of upper bounds on the size of EA-CWS codes, which is one of the main results in the paper (C-Y Lai, P-C Tseng, and W-H Yu 2023). Any improvement and discussion will be appreciated.

\centering \begin{threeparttable} \caption{The upper bounds on the size of EA-CWS codes} \label{table:sdp} \begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline \diagbox{$(n, c)$}{$d$} & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$\\ \hline $(3, 1)$ & $2$ & & & & & & & \\ \hline $(4, 1)$ & $2$ & $2$ & & & & & & \\ \hline $(5, 1)$ & $4$ & $2$ & $2$ & & & & & \\ \hline $(6, 1)$ & $5$ & $2$ & $2$ & $2$ & & & & \\ \hline $(7, 1)$ & $9$ & $2$ & $2$ & $2$ & $2$ & & & \\ \hline $(8, 1)$ & $18$ & $3$ & $2$ & $2$ & $2$ & $2$ & & \\ \hline $(9, 1)$ & $32$ & $5$ & $2$ & $2$ & $2$ & $2$ & $2$ & \\ \hline $(10, 1)$ & $58$ & $13$ & $2$ & $2$ & $2$ & $2$ & $2$ & $2$\\ \hline $(3, 2)$ & $3\ (4)$ & & & & & & & \\ \hline $(4, 2)$ & $4$ & $2\ (4)$ & & & & & & \\ \hline $(5, 2)$ & $8$ & $4$ & $2\ (4)$ & & & & & \\ \hline $(6, 2)$ & $11$ & $4$ & $4$ & $2\ (4)$ & & & & \\ \hline $(7, 2)$ & $19$ & $5$ & $4$ & $3\ (4)$ & $2\ (4)$ & & & \\ \hline $(8, 2)$ & $36$ & $8$ & $4$ & $4$ & $3\ (4)$ & $2\ (4)$ & & \\ \hline $(9, 2)$ & $67$ & $18$ & $4$ & $4$ & $4$ & $3\ (4)$ & $2\ (4)$ & \\ \hline $(10, 2)$ & $118$ & $31$ & $7$ & $4$ & $4$ & $4$ & $3\ (4)$ & $2\ (4)$\\ \hline $(4, 3)$ & $8$ & $3\ (8)$ & & & & & & \\ \hline $(5, 3)$ & $16$ & $6\ (8)$ & $3\ (8)$ & & & & & \\ \hline $(6, 3)$ & $22$ & $8$ & $5\ (8)$ & $3\ (8)$ & & & & \\ \hline $(7, 3)$ & $38$ & $11$ & $8$ & $4\ (8)$ & $2\ (8)$ & & & \\ \hline $(8, 3)$ & $73$ & $18$ & $8$ & $8$ & $4\ (8)$ & $2\ (8)$ & & \\ \hline $(9, 3)$ & $140$ & $36$ & $9$ & $8$ & $8$ & $4\ (8)$ & $2\ (8)$ & \\ \hline $(10, 3)$ & $237$ & $70$ & $16$ & $8$ & $8$ & $7\ (8)$ & $4\ (8)$ & $3\ (8)$\\ \hline $(5, 4)$ & $32$ & $10\ (16)$ & $3\ (16)$ & & & & & \\ \hline $(6, 4)$ & $44$ & $16$ & $7\ (16)$ & $3\ (16)$ & & & & \\ \hline $(7, 4)$ & $76$ & $22$ & $15\ (16)$ & $5\ (16)$ & $3\ (16)$ & & & \\ \hline $(8, 4)$ & $146$ & $38$ & $16$ & $12\ (16)$ & $5\ (16)$ & $3\ (16)$ & & \\ \hline $(9, 4)$ & $289$ & $72\ (73)$ & $19$ & $16$ & $10\ (16)$ & $4\ (16)$ & $3\ (16)$ & \\ \hline $(10, 4)$ & $475$ & $140\ (144)$ & $33$ & $16$ & $16$ & $9\ (10)$ & $4\ (16)$ & $3\ (16)$\\ \hline $(6, 5)$ & $89$ & $31\ (32)$ & $8\ (32)$ & $3\ (32)$ & & & & \\ \hline $(7, 5)$ & $153$ & $45$ & $20\ (32)$ & $6\ (32)$ & $3\ (32)$ & & & \\ \hline $(8, 5)$ & $292$ & $76$ & $32$ & $16\ (32)$ & $5\ (32)$ & $3\ (32)$ & & \\ \hline $(9, 5)$ & $585$ & $145\ (146)$ & $32$ & $32$ & $13\ (32)$ & $5\ (32)$ & $3\ (32)$ & \\ \hline $(7, 6)$ & $307$ & $90$ & $28\ (64)$ & $7\ (64)$ & $3\ (64)$ & & & \\ \hline $(8, 6)$ & $585$ & $153$ & $64$ & $20\ (64)$ & $6\ (64)$ & $3\ (64)$ & & \\ \hline $(9, 6)$ & $1170$ & $290\ (292)$ & $64$ & $51\ (64)$ & $17\ (64)$ & $5\ (64)$ & $3\ (64)$ & \\ \hline $(8, 7)$ & $1170$ & $306\ (307)$ & $112\ (128)$ & $26\ (128)$ & $6\ (128)$ & $3\ (128)$ & & \\ \hline $(9, 7)$ & $2340$ & $581\ (585)$ & $153$ & $74\ (128)$ & $22\ (128)$ & $6\ (128)$ & $3\ (128)$ & \\ \hline $(9, 8)$ & $4681$ & $1164\ (1170)$ & $307$ & $96\ (256)$ & $24\ (256)$ & $6\ (256)$ & $3\ (256)$ & \\ \hline \end{tabular} \begin{tablenotes} \item[2] The notation $a\ (b)$ indicates that $a$ is the result obtained through SDP, while $b$ is the result obtained through LP. If an entry contains only one value, it means that the SDP and LP bounds are identical. \end{tablenotes} \end{threeparttable} \section*{References} \bibliographystyle{alphaurl}
C-Y Lai, P-C Tseng, and W-H Yu. 2023. “Semidefinite Programming Bounds on the Size of Entanglement-Assisted Codeword Stabilized Quantum Codes.” https://arxiv.org/abs/2311.07111.
  • comment

No comment found.

Add a comment

You must sign in to post a comment.