OFFSET
1,2
COMMENTS
a(8) = 22, 23, or 24. a(9) = 24 or 25. a(10) = 28. a(11) = 29. 34 <= a(12) <= 42.
In 1995, Shor constructed the first true quantum error correcting code, a [[9,1,3]] code, showing a(3) <= 9.
LINKS
Charles Henry Bennett, David Peter DiVincenzo, John A. Smolin, and William Kent Wootters, Mixed-state entanglement and quantum error correction, Physical Review A 54.5 (1996), pp. 3824-3851.
A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over GF(4), arXiv:quant-ph/9608006, 1996-1997; IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.
Martianus Frederic Ezerman, Markus Grassl, San Ling, Ferruh Özbudak, and Buket Özkaya, Characterization of Nearly Self-Orthogonal Quasi-Twisted Codes and Related Quantum Codes, arXiv preprint (2024). arXiv:2405.15057 [quant-ph]
Markus Grassl, Searching for linear codes with large minimum distance. in: Wieb Bosma and John Cannon (Eds.), Discovering Mathematics with Magma, Springer, 2006.
Raymond Laflamme, Cesar Miquel, Juan Pablo Paz, and Wojciech Hubert Zurek, Perfect quantum error correcting code, Physical Review Letters 77.1 (1996), pp. 198-201.
Eric M. Rains, Nonbinary quantum codes, IEEE Transactions on Information Theory 45.6 (1999), pp. 1827-1832.
Eric M. Rains, Quantum shadow enumerators, IEEE transactions on information theory 45.7 (1999), pp. 2361-2366.
Peter W. Shor, Scheme for reducing decoherence in quantum computer memory, Physical review A 52.4 (1995), pp. R2493-R2496.
FORMULA
a(n) ~ 2n. a(n) >= 2n - 1 from the quantum Singleton bound.
EXAMPLE
a(1) = 1 from the trivial [[1,1,1]] code (no encoding).
a(2) = 4 from the [[4,1,2]] code, a subcode of the [[4,2,2]] error-detecting code.
a(3) = 5 from the perfect [[5,1,3]] stabilizer code (Laflamme, Miquel, Paz, & Zurek and Bennett, DiVincenzo, Smolin, & Wootters).
a(4) = 10 from a [[10,1,4]] code, a subcode of a [[10,2,4]] additive code over GF(4).
a(5) = 11 from a [[11,1,5]] additive code over GF(4).
a(6) = 16 from a [[16,1,6]] code, a subcode of a [[16,2,6]] additive code over GF(4).
a(7) = 17 from a [[17,1,7]] additive code over GF(4).
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Charles R Greathouse IV, Mar 11 2026
STATUS
approved