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

generates a code of length 77 over Z2[X]/(X^4) who´s minimum homogenous weight is 72.

Homogenous weight enumerator: w(x)=1x^0+164x^72+52x^74+574x^76+512x^77+488x^78+182x^80+4x^82+66x^84+4x^88+1x^144

The gray image is a linear code over GF(2) with n=616, k=11 and d=288.
This code was found by Heurico 1.16 in 0.656 seconds.