The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  2  1  1  1  1  1  1  1  1  1  1  1  1
 0  X  0 X^2+X+2  2 X^2+X  0  X X^2 X^2+X X^2+2  X X^2 X+2 X^2 X^2+X  0 X^2+X X^2 X+2  2 X^2+X+2  2 X^2+X  0 X^2+X X^2+2  X X^2+2  X X^2  X  0  X X^2 X+2  0 X+2 X^2 X^2+X+2  2  0 X^2+X  X X^2 X^2+2 X^2+X X^2+X+2 X^2+X+2  X X^2 X^2+2 X^2+X X+2 X^2+X  2  X X^2+X+2 X+2  2  0  0  2 X^2 X^2+2 X^2+2 X^2  0
 0  0 X^2+2  0  0 X^2+2 X^2 X^2 X^2  2 X^2+2  2  2 X^2  2 X^2+2  0  0  2 X^2+2 X^2+2 X^2 X^2  2  2  0  0 X^2 X^2 X^2+2 X^2+2  0  2  0  0 X^2+2 X^2 X^2+2 X^2  2 X^2  2 X^2+2  2 X^2+2  0 X^2  0  2 X^2 X^2 X^2+2 X^2  2 X^2  0  0 X^2+2  2  0 X^2+2  2 X^2+2 X^2+2  0  2 X^2+2  0
 0  0  0 X^2+2 X^2 X^2+2 X^2  0  0  0 X^2 X^2+2 X^2 X^2+2  0  0  2  2 X^2+2 X^2  2  2 X^2+2 X^2  0  0 X^2+2 X^2 X^2  2  0 X^2+2 X^2+2  2  0  0  2 X^2+2 X^2+2 X^2+2  0  2  2  2 X^2+2 X^2 X^2+2 X^2  2  2  2  2  0  0 X^2  0 X^2 X^2 X^2 X^2+2 X^2+2 X^2 X^2  2  2  2  2  0

generates a code of length 68 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 64.

Homogenous weight enumerator: w(x)=1x^0+55x^64+224x^65+72x^66+832x^67+48x^68+544x^69+15x^70+24x^72+192x^73+40x^74+1x^134

The gray image is a code over GF(2) with n=544, k=11 and d=256.
This code was found by Heurico 1.16 in 0.36 seconds.