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

generates a code of length 73 over Z2[X]/(X^4) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+346x^68+56x^69+534x^70+168x^71+911x^72+336x^73+720x^74+144x^75+466x^76+56x^77+182x^78+8x^79+131x^80+36x^82+1x^120

The gray image is a linear code over GF(2) with n=584, k=12 and d=272.
This code was found by Heurico 1.16 in 6.56 seconds.