SDP bounds on quantum codes

Maximum code size $K$ of nonadditive quantum codes for a given $n$ and $\delta$.
Lower and upper bounds are separated by a dash $-$.
Upper and lower bounds meet if only one number is shown.
A stronger SDP bound is highlighted in bold with the previous bound in parenthesis.

n $\backslash$ $\delta$	2	3	4	5	6	7	8
6	16	$2^\alpha$	1	0	0	0	0
7	24 - 26	2 - 3	0	0	0	0	0
8	64	8(9)	1	0	0	0	0
9	100 - 112	12 - 13	1	0	0	0	0
10	256	24	4(5)	0	0	0	0
11	416 - 460	32 - $\mathbf{42}^\alpha$(53)	4 - 7	2	0	0	0
12	1024	64 - 89	16 - 20	2	1	0	0
13	1586 - 1877	128 - 204	20 - 40	2 - 3	0(1)	0	0
14	4096	256 - $\mathbf{295}^\alpha$(324)	64 - 102	4 - $\mathbf{9}^\alpha$(10)	1	0	0
15	6476 - 7606	512 - 580	64 - $\mathbf{138}^\alpha$(150)	8 - 18	1	0	0
16	16384	1024 - 1170(1260)	128 - 256	32 - 42	4 - 6	0	0
17	26333 - 30720	2048 - 2145	512	32 - $\mathbf{59}^\alpha$(71)	4 - 8	2	0
18	65536	2048 - 4096	512 - 921(986)	64 - 113	16 - 22	2	1
19	106762 - 123790	4096 - $\mathbf{7420}^\alpha$(8426)	1024 - 1804	128 - 249(276)	32 - 47	2 - 3	0(1)

For pure codes the bounds marked with $\alpha$ strenghten to:

\[(\!(6,1,3)\!)_2 \,, \quad (\!(11,41,3)\!)_2 \,, \quad (\!(14,290,3)\!)_2 \,, (\!(14,8,5)\!)_2 \,, \quad (\! (15,135,4)\!)_2\,, \quad (\!(17,57,5)\!)_2 \,, \quad (\!(19,7314,3)\!)_2 \,.\]

Updates since 10 March 2026:

(none)

References

Upper bounds

are based on the SDP framework developed in

[MNH24] Gerard Anglès Munné, Andrew Nemec, Felix Huber, SDP bounds on quantum codes, arXiv:2408.10323

[MH46] Gerard Anglès Munné, Felix Huber, SDP bounds on quantum codes: rational certificates, arXiv:2603.19901

Lower bounds

[ROJ19] Alex Rigby, J. C. Olivier, and Peter Jarvis, Heuristic construction of codeword stabilized codes. Phys. Rev. A, 100(6), arXiv1907.04537.
[Gra07] Markus Grassl. Bounds on the minimum distance of linear codes and quantum codes, 2007. Online available at http://www.codetables.de.
[SSW07] John A. Smolin, Graeme Smith, and Stephanie Wehner. Simple family of nonadditive quantum codes. Phys. Rev. Lett. 99:130505, 2007.

Cite as: