The generator matrix

 1  0  1  1 X^2  1  1  1 X^2+X  1  1  0 X^3+X  1  1  1  1 X^2 X^3  1  1  1  1  1 X^3  1 X^3+X  1  1  1  1 X^3+X^2+X  1 X^2+X  1  1  1  1 X^2+X X^3+X  1  1  1  1  1 X^3+X^2+X  1  0  0 X^3+X^2+X X^3+X^2+X  X  1  1  1  1  1  1  1  1 X^3+X  1 X^3+X^2+X X^2  1 X^3+X^2+X  1  1  0  X  1  1  X  1  1  1
 0  1  1 X^2+X  1 X^2+X+1 X^2 X^3+1  1 X+1 X^3+X^2+X  1  1  0 X^3+X^2+1 X^3 X^3+1  1  1  X X+1 X^2+X X^3+X+1 X^3+X^2  1  1  1 X^2 X^3 X^2+1 X^3+1  1 X^3+X^2+X+1  1 X^3+X X^2+X+1 X^3 X+1  1  1 X+1  X X^2+1  X  X  1 X+1  1  1  1  1 X^3+X^2+X X^2+X X^2+1 X^2+X+1 X^3+1 X^3+1 X^3+X^2+X X+1 X^3+X  1 X^3+X^2+X  1  1 X^2+1  1 X^3+X^2+X+1 X^2+X+1  1  1  X  1  X X^3+X+1 X^3+1  X
 0  0  X  0 X^3+X  X X^3+X X^3  0 X^3 X^3+X X^3+X^2+X X^2 X^3+X^2 X^3+X^2 X^3+X^2+X X^3+X^2+X X^2+X X^3+X X^2 X^2+X X^2+X X^2 X^2+X X^3+X^2 X^2 X^3+X^2+X X^2 X^3+X X^3  X X^3+X^2 X^3+X^2 X^3+X^2+X X^3+X^2+X  X  0 X^3+X^2  X  0 X^3+X^2+X X^3+X^2 X^3+X^2+X  0 X^3+X X^3+X X^3  0 X^3+X^2 X^3 X^3+X^2 X^2+X X^2  X X^2+X X^2+X X^3 X^3  0 X^3+X^2+X X^3+X X^3+X X^3+X^2+X X^3+X^2 X^3+X X^2 X^2+X X^3+X^2 X^3+X^2+X  X  X X^3+X^2 X^3+X^2+X X^2+X X^2 X^3+X^2+X
 0  0  0 X^3  0 X^3 X^3 X^3 X^3  0  0 X^3 X^3  0 X^3 X^3  0  0 X^3 X^3 X^3  0  0  0  0  0 X^3 X^3  0  0 X^3  0  0  0  0  0 X^3 X^3 X^3 X^3  0  0 X^3  0 X^3  0 X^3 X^3 X^3  0 X^3  0  0  0  0 X^3  0  0 X^3 X^3  0 X^3 X^3  0 X^3  0 X^3 X^3  0 X^3  0  0 X^3  0 X^3  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+382x^72+512x^73+624x^74+408x^75+512x^76+424x^77+372x^78+360x^79+296x^80+88x^81+64x^82+18x^84+28x^86+4x^88+1x^92+1x^100+1x^104

The gray image is a linear code over GF(2) with n=608, k=12 and d=288.
This code was found by Heurico 1.16 in 0.656 seconds.