Table of Lower and Upper Bounds on Minimum Distance d of q-ary Stabilizer Quantum-Error-Correcting Codes

We give here some preliminary results on the bounds. The upper bounds are derived by linear programming. The lower bounds are obtained in a constructive way, mostly through quantum BCH code constructions.
We will post improved tables later this summer.

 

n/k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
3 2-2 1-1 1-1 - - - - - - - - - - - - - - - - - - - - -
4 2-2 2-2 1-1 1-1 - - - - - - - - - - - - - - - - - - - -
5 3-3 2-2 2-2 1-1 1-1 - - - - - - - - - - - - - - - - - - -
6 3-3 3-3 2-2 2-2 1-1 1-1 - - - - - - - - - - - - - - - - - -
7 3-4 3-3 3-3 2-2 2-2 1-1 1-1 - - - - - -   - - - - - - - - - -
8 3-4 3-4 3-3 3-3 2-2 2-2 1-1 1-1 - - - - - - - - - - - - - - - -
9 4-5 3-4 3-4 3-3 3-3 2-2 2-2 1-1 1-1 - - - - - - - - - - - - - - -
10 4-5 3-5 3-4 3-4 3-3 3-3 2-2 2-2 1-1 1-1 - - - - - - - - - - - - - -
11 4-6 3-5 3-5 3-4 3-4 3-3 2-2 2-2 1-2 1-1 1-1 - - - - - - - - - - - - -
12 4-6 4-6 3-5 3-5 3-4 3-3 2-3 2-2 1-2 1-2 1-1 1-1 - - - - - - - - - - - -
13 4-7 4-6 4-6 3-5 3-4 3-4 2-3 2-3 2-2 1-2 1-2 1-1 1-1 - - - - - - - - - - -
14 5-7 5-7 4-6 3-5 3-5 3-4 3-4 3-3 2-3 2-2 1-2 1-2 1-1 1-1 - - - - - - - - - -
15 6-8 5-7 4-6 3-6 3-5 3-5 3-4 3-4 2-3 2-3 2-2 1-2 1-2 1-1 1-1 - - - - - - - - -
16 6-8 5-7 4-7 4-6 3-6 3-5 3-5 3-4 3-4 3-3 2-3 2-2 1-2 1-2 1-1 1-1 - - - - - - - -
17 6-8 5-8 5-7 4-7 3-6 3-6 3-5 3-5 3-4 3-4 2-3 2-3 2-2 1-2 1-2 1-1 1-1 - - - - - - -
18 6-9 6-8 5-8 4-7 4-7 4-6 4-6 3-5 3-5 3-4 3-4 3-3 2-3 2-2 1-2 1-2 1-1 1-1 - - - - - -
19 6-9 6-9 5-8 4-8 4-7 4-7 4-6 3-6 3-5 3-5 3-4 3-4 3-3 2-3 2-2 1-2 1-2 1-1 1-1 - - - - -
20 6-10 6-9 5-9 5-8 4-8 4-7 4-7 4-6 4-6 4-5 4-5 4-4 3-4 3-3 2-3 2-2 1-2 1-2 1-1 1-1 - - - -
21 6-10 6-10 6-9 5-9 5-8 5-8 5-7 5-7 5-6 5-6 5-5 4-5 3-4 3-4 3-3 2-3 1-2 1-2 1-2 1-1 1-1 - - -
22 6-11 6-10 6-10 6-9 6-9 6-8 6-8 6-7 6-7 6-6 5-6 4-5 4-5 4-4 3-4 2-3 1-3 1-2 1-2 1-2 1-1 1-1 - -
23 7-11 7-11 7-10 7-10 7-9 7-9 7-8 7-8 7-7 6-7 5-6 5-6 5-5 4-5 3-4 2-4 1-3 1-3 1-2 1-2 1-2 1-1 1-1 -
24 8-11 8-11 8-11 8-10 8-10 8-9 8-9 8-8 7-8 6-7 6-7 6-6 5-5 4-5 3-4 2-4 1-4 1-3 1-3 1-2 1-2 1-2 1-1 1-1

Table 1:     Lower and Upper Bounds on Minimum Distance d of Ternary Stabilizer Quantum-Error-Correcting Codes

 

n/k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
3 2-2 1-1 1-1 - - - - - - - - - - - - - - - - - - - - -
4 2-2 2-2 1-1 1-1 - - - - - - - - - - - - - - - - - - - -
5 3-3 2-2 2-2 1-1 1-1 - - - - - - - - - - - - - - - - - - -
6 3-3 3-3 2-2 2-2 1-1 1-1 - - - - - - - - - - - - - - - - - -
7 3-4 3-3 3-3 2-2 2-2 1-1 1-1 - - - - - - - - - - - - - - - - -
8 4-4 4-4 3-3 3-3 2-2 2-2 1-1 1-1 - - - - - - - - - - - - - - - -
9 4-5 4-4 3-4 3-3 3-3 2-2 2-2 1-1 1-1 - - - - - - - - - - - - - - -
10 4-5 4-5 4-4 4-4 3-3 3-3 2-2 2-2 1-1 1-1 - - - - - - - - - - - - - -
11 4-6 4-5 4-5 4-4 4-4 3-3 3-3 2-2 2-2 1-1 1-1 - - - - - - - - - - - - -
12 4-6 4-6 4-5 4-5 4-4 4-4 3-3 3-3 2-2 2-2 1-1 1-1 - - - - - - - - - - - -
13 4-7 4-6 4-6 4-5 4-5 4-4 3-4 3-3 3-3 2-2 2-2 1-1 1-1 - - - - - - - - - - -
14 5-7 5-7 4-6 4-6 4-5 4-5 4-4 4-4 3-3 3-3 2-2 2-2 1-1 1-1 - - - - - - - - - -
15 6-8 5-7 4-7 4-6 4-6 4-5 4-5 4-4 4-4 3-3 3-3 2-2 2-2 1-1 1-1 - - - - - - - - -
16 6-8 5-8 4-7 4-7 4-6 4-6 4-5 4-5 4-4 4-4 3-3 3-3 2-2 2-2 1-1 1-1 - - - - - - - -
17 6-9 5-8 5-8 4-7 4-7 4-6 4-6 4-5 4-5 4-4 3-4 3-3 2-3 2-2 1-2 1-1 1-1 - - - - - - -
18 6-9 6-9 5-8 4-8 4-7 4-7 4-6 4-6 4-5 4-5 3-4 3-4 2-3 2-2 2-2 2-2 1-1 1-1 - - - - - -
19 6-10 6-9 5-9 4-8 4-8 4-7 4-7 3-6 4-6 4-5 3-5 3-4 3-3 3-3 2-2 1-2 1-2 1-1 1-1 - - - - -
20 6-10 6-10 5-9 5-9 4-8 4-8 4-7 4-7 4-6 4-6 4-5 4-4 4-4 3-3 2-3 2-2 2-2 2-2 1-1 1-1 - - - -
21 6-11 6-10 6-10 5-9 5-9 4-8 4-8 4-7 4-7 4-6 4-5 4-5 4-4 2-4 2-3 2-3 2-2 1-2 1-2 1-1 1-1 - - -
22 6-11 6-11 6-10 6-10 5-9 4-9 4-8 4-8 4-7 4-6 4-6 4-5 4-5 3-4 3-4 3-3 2-3 2-2 2-2 2-2 1-1 1-1 - -
23 7-12 7-11 7-11 6-10 5-10 4-9 4-9 4-8 4-7 4-7 4-6 4-6 4-5 3-5 3-4 3-4 3-3 3-3 2-3 2-2 1-2 1-1 1-1 -
24 8-12 8-12 7-11 6-11 5-10 4-10 4-9 4-8 4-8 4-7 4-7 4-6 4-6 4-5 4-5 4-4 4-4 3-3 2-3 2-2 2-2 2-2 1-1 1-1

Table 2:     Lower and Upper Bounds on Minimum Distance d of Quaternary Stabilizer Quantum-Error-Correcting Codes