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

generates a code of length 77 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 72.

Homogenous weight enumerator: w(x)=1x^0+146x^72+64x^74+342x^76+1024x^77+256x^78+138x^80+58x^84+18x^88+1x^144

The gray image is a code over GF(2) with n=616, k=11 and d=288.
This code was found by Heurico 1.16 in 3.03 seconds.