Encoding Circuits for Quantum Error-Correcting Codes

(notes)

Codes up to length 30 encoding up to 11 qubits

n/k 0 1 2 3 4 5 6 7 8 9 10 11
2 [[2,0,2]]                      
3 [[3,0,2]]                      
4 [[4,0,2]] [[4,1,2]] [[4,2,2]]                  
5 [[5,0,3]] [[5,1,3]] [[5,2,2]]
((5,6,2))		                  
6 [[6,0,4]] [[6,1,3]] [[6,2,2]] [[6,3,2]] [[6,4,2]]              
7 [[7,0,3]] [[7,1,3]] [[7,2,2]] [[7,3,2]] [[7,4,2]]              
8 [[8,0,4]] [[8,1,3]] [[8,2,3]] [[8,3,3]] [[8,4,2]] [[8,5,2]] [[8,6,2]]          
9 [[9,0,4]] [[9,1,3]] [[9,2,3]] [[9,3,3]]
((9,12,3))		 [[9,4,2]] [[9,5,2]] [[9,6,2]]          
10 [[10,0,4]] [[10,1,4]] [[10,2,4]] [[10,3,3]] [[10,4,3]]
((10,24,3))		 [[10,5,2]] [[10,6,2]] [[10,7,2]] [[10,8,2]]      
11 [[11,0,5]] [[11,1,5]] [[11,2,4]] [[11,3,3]] [[11,4,3]] [[11,5,3]] [[11,6,2]] [[11,7,2]] [[11,8,2]]      
12 [[12,0,6]] [[12,1,5]] [[12,2,4]] [[12,3,4]] [[12,4,4]] [[12,5,3]] [[12,6,3]] [[12,7,2]] [[12,8,2]] [[12,9,2]] [[12,10,2]]  
13 [[13,0,5]] [[13,1,5]] [[13,2,4]] [[13,3,4]] [[13,4,4]] [[13,5,3]] [[13,6,3]] [[13,7,3]] [[13,8,2]] [[13,9,2]] [[13,10,2]]  
14 [[14,0,6]] [[14,1,5]] [[14,2,5]] [[14,3,4]] [[14,4,4]] [[14,5,4]] [[14,6,4]] [[14,7,3]] [[14,8,3]] [[14,9,2]] [[14,10,2]] [[14,11,2]]
15 [[15,0,6]] [[15,1,5]] [[15,2,5]] [[15,3,5]] [[15,4,4]] [[15,5,4]] [[15,6,4]] [[15,7,3]] [[15,8,3]] [[15,9,3]] [[15,10,2]] [[15,11,2]]
16 [[16,0,6]] [[16,1,6]] [[16,2,6]] [[16,3,5]] [[16,4,5]] [[16,5,4]] [[16,6,4]] [[16,7,4]] [[16,8,3]] [[16,9,3]] [[16,10,3]] [[16,11,2]]
17 [[17,0,7]] [[17,1,7]] [[17,2,6]] [[17,3,5]] [[17,4,5]] [[17,5,4]] [[17,6,4]] [[17,7,4]] [[17,8,4]] [[17,9,4]] [[17,10,3]] [[17,11,3]]
18 [[18,0,8]] [[18,1,7]] [[18,2,6]] [[18,3,5]] [[18,4,5]] [[18,5,5]] [[18,6,5]] [[18,7,4]] [[18,8,4]] [[18,9,4]] [[18,10,3]] [[18,11,3]]
19 [[19,0,7]] [[19,1,7]] [[19,2,6]] [[19,3,5]] [[19,4,5]] [[19,5,5]] [[19,6,5]] [[19,7,4]] [[19,8,4]] [[19,9,4]] [[19,10,3]] [[19,11,3]]
20 [[20,0,8]] [[20,1,7]] [[20,2,6]] [[20,3,6]] [[20,4,6]] [[20,5,5]] [[20,6,5]] [[20,7,4]] [[20,8,4]] [[20,9,4]] [[20,10,4]] [[20,11,3]]
21 [[21,0,8]] [[21,1,7]] [[21,2,6]] [[21,3,6]] [[21,4,6]] [[21,5,6]] [[21,6,5]] [[21,7,5]] [[21,8,4]] [[21,9,4]] [[21,10,4]] [[21,11,4]]
22 [[22,0,8]] [[22,1,7]] [[22,2,6]] [[22,3,6]] [[22,4,6]] [[22,5,6]] [[22,6,5]] [[22,7,5]] [[22,8,5]] [[22,9,4]] [[22,10,4]] [[22,11,4]]
23 [[23,0,8]] [[23,1,7]] [[23,2,7]] [[23,3,6]] [[23,4,6]] [[23,5,6]] [[23,6,5]] [[23,7,5]] [[23,8,5]] [[23,9,4]] [[23,10,4]] [[23,11,4]]
24 [[24,0,8]] [[24,1,8]] [[24,2,7]] [[24,3,7]] [[24,4,6]] [[24,5,6]] [[24,6,6]] [[24,7,5]] [[24,8,5]] [[24,9,5]] [[24,10,5]] [[24,11,4]]
25 [[25,0,8]] [[25,1,9]] [[25,2,7]] [[25,3,7]] [[25,4,7]] [[25,5,7]] [[25,6,6]] [[25,7,5]] [[25,8,5]] [[25,9,5]] [[25,10,5]] [[25,11,4]]
26 [[26,0,8]] [[26,1,9]] [[26,2,8]] [[26,3,8]] [[26,4,8]] [[26,5,7]] [[26,6,6]] [[26,7,6]] [[26,8,6]] [[26,9,5]] [[26,10,5]] [[26,11,4]]
27 [[27,0,9]] [[27,1,9]] [[27,2,9]] [[27,3,9]] [[27,4,8]] [[27,5,7]] [[27,6,6]] [[27,7,6]] [[27,8,6]] [[27,9,6]] [[27,10,5]] [[27,11,5]]
28 [[28,0,10]] [[28,1,10]] [[28,2,10]] [[28,3,9]] [[28,4,8]] [[28,5,7]] [[28,6,6]] [[28,7,6]] [[28,8,6]] [[28,9,6]] [[28,10,6]] [[28,11,5]]
29 [[29,0,11]] [[29,1,11]] [[29,2,10]] [[29,3,9]] [[29,4,8]] [[29,5,7]] [[29,6,7]] [[29,7,6]] [[29,8,6]] [[29,9,6]] [[29,10,6]] [[29,11,5]]
30 [[30,0,12]] [[30,1,11]] [[30,2,10]] [[30,3,9]] [[30,4,8]] [[30,5,8]] [[30,6,7]] [[30,7,7]] [[30,8,7]] [[30,9,6]] [[30,10,6]] [[30,11,6]]
k maximal for fixed (n,d) and
n minimal for fixed (k,d) 		better non-additive code exists d maximal for fixed (n,k) d=1 k>n

Other Codes

[[23,12,4]] [[25,15,4]] [[28,14,5]] [[30,20,4]] [[40,30,4]]

Notes

Copyright © Markus Grassl ( grassl@ira.uka.de) 13.09.2007