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  X  1  1  1  1  X  X  1  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  X X^2  X  X  X  X  X  1
 0 X^3+X^2  0  0  0 X^2 X^3+X^2 X^2  0  0  0  0 X^2 X^3+X^2 X^2 X^3+X^2  0  0 X^2 X^3+X^2  0 X^2  0 X^3+X^2 X^3 X^3+X^2 X^3  0 X^3+X^2  0 X^2 X^3 X^3  0 X^2 X^3  0 X^3+X^2 X^3 X^2 X^3 X^3+X^2 X^3 X^2 X^3 X^3+X^2 X^3 X^2 X^3 X^3+X^2 X^3 X^2 X^3 X^3+X^2 X^3 X^2 X^3 X^3+X^2 X^3 X^2 X^3 X^3+X^2 X^3 X^2 X^3 X^3 X^3+X^2 X^2 X^2 X^3 X^3+X^2 X^3+X^2 X^3+X^2  0 X^3  0
 0  0 X^3+X^2  0 X^2 X^2 X^3+X^2  0  0  0 X^2 X^3+X^2 X^2 X^3+X^2  0  0 X^3 X^3 X^3+X^2 X^2 X^3+X^2 X^3 X^2 X^3 X^3 X^3+X^2 X^2 X^3 X^2 X^3+X^2 X^3  0 X^3 X^3 X^3+X^2 X^2 X^2 X^3  0 X^2 X^3+X^2  0 X^3+X^2  0 X^3 X^3+X^2 X^2 X^3 X^2 X^3  0 X^2  0 X^2 X^2 X^3+X^2 X^2 X^3 X^3 X^3  0 X^2 X^3 X^3+X^2 X^3+X^2 X^3+X^2  0  0 X^2 X^3+X^2 X^2 X^3 X^3+X^2  0 X^3  0
 0  0  0 X^3+X^2 X^2  0 X^3+X^2 X^2 X^3 X^2 X^3+X^2 X^3 X^3 X^2 X^3+X^2 X^3 X^3 X^2 X^3 X^2 X^3+X^2 X^3+X^2 X^3 X^3  0  0  0 X^3+X^2 X^3+X^2 X^2 X^2 X^2 X^2  0  0 X^3  0  0 X^2 X^3+X^2 X^2 X^2 X^3  0 X^3 X^3  0  0 X^3+X^2 X^2 X^3+X^2 X^2 X^3 X^3 X^3 X^2 X^2 X^3+X^2 X^3+X^2 X^3  0  0 X^2 X^3+X^2 X^3+X^2  0 X^3+X^2 X^3  0 X^2 X^2 X^3+X^2 X^3+X^2  0  0  0

generates a code of length 76 over Z2[X]/(X^4) who´s minimum homogenous weight is 72.

Homogenous weight enumerator: w(x)=1x^0+21x^72+72x^73+43x^74+212x^75+294x^76+278x^77+18x^78+60x^79+4x^80+16x^81+2x^82+2x^85+1x^130

The gray image is a linear code over GF(2) with n=608, k=10 and d=288.
This code was found by Heurico 1.16 in 0.36 seconds.