The generator matrix

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

generates a code of length 39 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 34.

Homogenous weight enumerator: w(x)=1x^0+80x^34+28x^35+147x^36+56x^37+452x^38+608x^39+417x^40+40x^41+78x^42+36x^43+60x^44+24x^46+14x^48+6x^50+1x^60

The gray image is a code over GF(2) with n=312, k=11 and d=136.
This code was found by Heurico 1.16 in 0.125 seconds.