The generator matrix

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

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+254x^72+64x^73+256x^74+64x^75+768x^76+64x^77+256x^78+64x^79+254x^80+1x^88+1x^104+1x^112

The gray image is a linear code over GF(2) with n=608, k=11 and d=288.
This code was found by Heurico 1.16 in 0.344 seconds.