Code details

best found code with parameters
q=4 k=5 n=56
minimum distance = 40

this is new optimal code


the previous bounds were 39/40
this is a projective code


We used the prescribed group of automorphisms with the following generators


3 0 0 0 0
0 3 0 0 0
0 0 3 0 0
0 0 0 3 0
0 0 0 0 3

0 0 0 0 2
2 0 0 0 0
0 2 0 0 0
0 0 2 0 0
0 0 0 2 0

This group makes 69 orbits of sizes:

5 5 5 5 5 5 5 5 5 1 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5


The solution of the corresponding linear system of equations was found after 12 minutes:

0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 12 16 12 8 16 16 16 12 0 16 12 16 16 16 16 8 16 12 16 8 16 16 16 8 16 16 16 16 16 16 16 16 8 12 16 16 16 16 12 16 12 16 16 8 12 16 12 16 8 16 16 12 16 16 16 16 12 12 8 16 12 16 16 16 16 16 12 8


This produces the following generator matrix

3 3 3 3 3 3 0 3 3 3 3 0 3 3 3 3 3 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 0 3 3 3 0 3 3 3 3 0 3 3 3 3 0 0 3 3 3 0 0 3 3 3
1 2 2 2 2 3 3 0 1 3 3 3 0 2 3 3 1 1 2 2 3 3 0 1 1 3 3 0 1 2 3 3 3 0 0 3 3 0 2 2 3 3 0 1 2 2 0 3 0 1 2 0 3 0 1 2
3 1 1 1 3 3 3 2 0 1 3 2 1 2 0 3 1 3 3 3 2 3 1 0 2 1 1 3 3 2 0 0 3 3 3 0 2 2 1 2 0 2 1 0 1 3 3 2 0 0 3 3 1 0 3 0
2 2 3 3 2 3 3 2 3 0 1 2 3 2 1 0 3 1 2 3 3 1 1 2 0 2 3 1 3 0 3 3 0 0 3 3 1 1 1 0 2 1 3 2 2 0 2 3 3 0 0 1 3 3 0 0
1 1 1 2 1 3 1 2 3 3 0 2 3 0 3 1 1 1 3 2 2 2 2 2 3 0 3 3 0 2 1 3 3 3 0 0 1 3 0 1 1 2 2 1 0 1 3 0 2 1 0 3 0 1 0 2



Which is a code with the following weight distribution
1y56+645x40y16+240x44y12+135x48y8+3x56