Encoding Circuits for Cyclic Quantum Error-Correcting Codes

Cyclic codes up to length 41 encoding up to 32 qubits

n/k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
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]] - -                                                          
6 [[6,0,4]] [[6,1,2]] [[6,2,2]] [[6,3,2]] [[6,4,2]]                                                        
7 [[7,0,3]] [[7,1,3]] - [[7,3,2]] [[7,4,2]] -                                                      
8 [[8,0,4]] [[8,1,3]] [[8,2,2]] [[8,3,2]] [[8,4,2]] [[8,5,2]] [[8,6,2]]                                                    
9 [[9,0,4]] [[9,1,3]] [[9,2,3]] [[9,3,3]] - - [[9,6,2]]                                                    
10 [[10,0,4]] [[10,1,4]] [[10,2,3]] - [[10,4,2]] [[10,5,2]] [[10,6,2]] - [[10,8,2]]                                                
11 [[11,0,4]] [[11,1,3]] - - - - - - - -                                              
12 [[12,0,6]] [[12,1,4]] [[12,2,4]] [[12,3,4]] [[12,4,3]] [[12,5,2]] [[12,6,2]] [[12,7,2]] [[12,8,2]] [[12,9,2]] [[12,10,2]]                                            
13 [[13,0,5]] [[13,1,5]] - - - - - - - - - -                                          
14 [[14,0,6]] [[14,1,5]] [[14,2,5]] [[14,3,4]] [[14,4,4]] [[14,5,3]] [[14,6,4]] [[14,7,3]] [[14,8,3]] [[14,9,2]] [[14,10,2]] [[14,11,2]] [[14,12,2]]                                        
15 [[15,0,6]] [[15,1,5]] [[15,2,5]] [[15,3,5]] [[15,4,4]] [[15,5,4]] [[15,4,4]] [[15,7,3]] [[15,8,3]] [[15,9,3]] [[15,10,2]] [[15,11,2]] [[15,12,2]]                                        
16 [[16,0,6]] [[16,1,6]] [[16,2,4]] [[16,3,4]] [[16,4,4]] [[16,5,4]] [[16,6,2]] [[16,7,2]] [[16,8,2]] [[16,9,2]] [[16,10,2]] [[16,11,2]] [[16,12,2]] [[16,13,2]] [[16,14,2]]                                    
17 [[17,0,7]] [[17,1,7]] - - - - - - [[17,8,4]] [[17,9,4]] - - - - - -                                  
18 [[18,0,6]] [[18,1,6]] [[18,2,6]] [[18,3,5]] [[18,4,5]] [[18,5,3]] [[18,6,4]] [[18,7,2]] [[18,8,2]] [[18,9,2]] [[18,10,2]] [[18,11,2]] [[18,12,2]] [[18,13,2]] [[18,14,2]] [[18,15,2]] [[18,16,2]]                                
19 [[19,0,7]] [[19,1,7]] - - - - - - - - - - - - - - - -                              
20 [[20,0,8]] [[20,1,6]] [[20,2,6]] [[20,3,5]] [[20,4,5]] [[20,5,4]] [[20,6,4]] [[20,7,4]] [[20,8,4]] [[20,9,2]] [[20,10,2]] [[20,11,2]] [[20,12,2]] [[20,13,2]] [[20,14,2]] [[20,15,2]] [[20,16,2]] [[20,17,2]] [[20,18,2]]                            
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]] [[21,12,3]] [[21,13,3]] [[21,14,3]] [[21,15,3]] [[21,16,2]] [[21,17,2]] [[21,18,2]]                            
22 [[22,0,8]] [[22,1,7]] [[22,2,6]] - - - - - - - [[22,10,2]] [[22,11,2]] [[22,12,2]] - - - - - - - [[22,20,2]]                        
23 [[23,0,8]] [[23,1,7]] - - - - - - - - - [[23,11,4]] [[23,12,4]] - - - - - - - - -                      
24 [[24,0,8]] [[24,1,8]] [[24,2,7]] [[24,3,7]] [[24,4,6]] [[24,5,6]] [[24,6,6]] [[24,7,4]] [[24,8,4]] [[24,9,4]] [[24,10,4]] [[24,11,4]] [[24,12,4]] [[24,13,4]] [[24,14,3]] [[24,15,3]] [[24,16,2]] [[24,17,2]] [[24,18,2]] [[24,19,2]] [[24,20,2]] [[24,21,2]] [[24,22,2]]                    
25 [[25,0,8]] [[25,1,9]] - - [[25,4,5]] [[25,5,5]] - - - - - - - - - - - - - - [[25,20,2]] [[25,21,2]] - -                  
26 [[26,0,8]] [[26,1,8]] [[26,2,8]] - - - - - - - - - [[26,12,2]] [[26,13,2]] [[26,14,2]] - - - - - - - - - [[26,24,2]]                
27 [[27,0,8]] [[27,1,9]] [[27,2,8]] [[27,3,7]] - - [[27,6,6]] [[27,7,3]] [[27,8,3]] - - - - - - - - - [[27,18,2]] [[27,19,2]] [[27,20,2]] [[27,21,2]] - - [[27,24,2]]                
28 [[28,0,8]] [[28,1,9]] [[28,2,8]] [[28,3,8]] [[28,4,7]] [[28,5,6]] [[28,6,6]] [[28,7,6]] [[28,8,6]] [[28,9,6]] [[28,10,6]] [[28,11,5]] [[28,12,4]] [[28,13,4]] [[28,14,4]] [[28,15,4]] [[28,16,4]] [[28,17,3]] [[28,18,2]] [[28,19,2]] [[28,20,2]] [[28,21,2]] [[28,22,2]] [[28,23,2]] [[28,24,2]] [[28,25,2]] [[28,26,2]]            
29 [[29,0,9]] [[29,1,11]] - - - - - - - - - - - - - - - - - - - - - - - - - -          
30 [[30,0,12]] [[30,1,8]] [[30,2,8]] [[30,3,8]] [[30,4,8]] [[30,5,7]] [[30,6,7]] [[30,7,6]] [[30,8,7]] [[30,9,6]] [[30,10,6]] [[30,11,6]] [[30,12,4]] [[30,13,4]] [[30,14,4]] [[30,15,4]] [[30,16,4]] [[30,17,4]] [[30,18,4]] [[30,19,3]] [[30,20,4]] [[30,21,3]] [[30,22,2]] [[30,23,2]] [[30,24,2]] [[30,25,2]] [[30,26,2]] [[30,27,2]] [[30,28,2]]        
31 [[31,0,10]] [[31,1,9]] - - - [[31,5,8]] [[31,6,7]] - - - [[31,10,6]] [[31,11,6]] - - - [[31,15,5]] [[31,16,5]] - - - [[31,20,4]] [[31,21,4]] - - - [[31,25,2]] [[31,26,2]] - - -      
32 [[32,0,10]] [[32,1,8]] [[32,2,8]] [[32,3,8]] [[32,4,8]] [[32,5,8]] [[32,6,4]] [[32,7,4]] [[32,8,4]] [[32,9,4]] [[32,10,4]] [[32,11,4]] [[32,12,4]] [[32,13,4]] [[32,14,2]] [[32,15,2]] [[32,16,2]] [[32,17,2]] [[32,18,2]] [[32,19,2]] [[32,20,2]] [[32,21,2]] [[32,22,2]] [[32,23,2]] [[32,24,2]] [[32,25,2]] [[32,26,2]] [[32,27,2]] [[32,28,2]] [[32,29,2]] [[32,30,2]]    
33 [[33,0,10]] [[33,1,9]] [[33,2,9]] [[33,3,9]] - - - - - - [[33,10,6]] [[33,11,6]] [[33,12,6]] [[33,13,5]] - - - - - - [[33,20,3]] [[33,21,3]] [[33,22,3]] [[33,23,3]] - - - - - - [[33,30,2]]    
34 [[34,0,10]] [[34,1,9]] [[34,2,8]] - - - - - [[34,8,8]] [[34,9,7]] [[34,10,6]] - - - - - [[34,16,6]] [[34,17,5]] [[34,18,5]] - - - - - [[34,24,2]] [[34,25,2]] [[34,26,2]] - - - - - [[34,32,2]]
35 [[35,0,10]] [[35,1,10]] - [[35,3,8]] [[35,4,8]] [[35,5,7]] [[35,6,5]] [[35,7,7]] [[35,8,7]] - [[35,10,5]] [[35,11,5]] [[35,12,6]] [[35,13,7]] - [[35,15,5]] [[35,16,6]] [[35,17,5]] [[35,18,4]] [[35,19,4]] [[35,20,5]] - [[35,22,3]] [[35,23,3]] [[35,24,4]] [[35,25,4]] - [[35,27,3]] [[35,28,3]] [[35,29,2]] [[35,30,2]] [[35,31,2]] [[35,32,2]]
for n>35 under construction
36 [[36,0,11]] [[36,1,10]] [[36,2,8]] [[36,3,8]] [[36,4,8]] [[36,5,6]] [[36,6,6]] [[36,7,6]] [[36,8,6]] [[36,9,6]] [[36,10,6]] [[36,11,4]] [[36,12,4]] [[36,13,4]] [[36,14,4]] [[36,15,4]] [[36,16,4]] [[36,17,3]] [[36,18,4]] [[36,19,4]] [[36,20,4]] [[36,21,4]] [[36,22,4]] [[36,23,2]] [[36,24,2]] [[36,25,2]] [[36,26,2]] [[36,27,2]] [[36,28,2]] [[36,29,2]] [[36,30,2]] [[36,31,2]] [[36,32,2]]
37 [[37,0,11]] [[37,1,11]] - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
38 [[38,0,11]] [[38,1,10]] [[38,2,9]] - - - - - - - - - - - - - - - [[38,18,2]] [[38,19,2]] - - - - - - - - - - - - -
39 [[39,0,10]] [[39,1,11]] [[39,2,10]] [[39,3,9]] - - - - - - - - [[39,12,7]] [[39,13,7]] [[39,14,6]] [[39,15,6]] - - - - - - - - [[39,24,3]] [[39,25,3]] [[39,26,3]] [[39,27,3]] - - - - -
40 [[40,0,11]] [[40,1,10]] [[40,2,10]] [[40,3,10]] [[40,4,10]] [[40,5,8]] [[40,6,8]] [[40,7,8]] [[40,8,6]] [[40,9,6]] [[40,10,6]] [[40,11,5]] [[40,12,4]] [[40,13,4]] [[40,14,4]] [[40,15,4]] [[40,16,4]] [[40,17,4]] [[40,18,4]] [[40,19,4]] [[40,20,4]] [[40,21,4]] [[40,22,]] [[40,23,]] [[40,24,4]] [[40,25,4]] [[40,26,]] [[40,27,]] [[40,28,]] [[40,29,]] [[40,30,]] [[40,31,]] [[40,32,]]
41 [[41,0,11]] [[41,1,11]] - - - - - - - - - - - - - - - - - - [[41,20,6]] [[41,21,6]] - - - - - - - - - - -
k maximal for fixed (n,d) and
n minimal for fixed (k,d) 		cyclic code with
larger k and d exists		 d maximal for
fixed n and k 		d=1 k>n
 

Notes

Copyright © Markus Grassl ( grassl@ira.uka.de) 05.02.2003