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

generates a code of length 75 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 72.

Homogenous weight enumerator: w(x)=1x^0+114x^72+256x^74+256x^75+310x^76+64x^78+20x^80+2x^84+1x^128

The gray image is a code over GF(2) with n=600, k=10 and d=288.
This code was found by Heurico 1.16 in 19 seconds.