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

generates a code of length 69 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 67.

Homogenous weight enumerator: w(x)=1x^0+72x^67+193x^68+496x^69+184x^70+72x^71+5x^72+1x^132

The gray image is a code over GF(2) with n=552, k=10 and d=268.
This code was found by Heurico 1.16 in 6.02 seconds.