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

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+35x^68+92x^69+178x^70+280x^71+319x^72+320x^73+423x^74+160x^75+13x^76+100x^77+34x^78+64x^79+16x^80+4x^82+8x^83+1x^138

The gray image is a linear code over GF(2) with n=584, k=11 and d=272.
This code was found by Heurico 1.16 in 0.609 seconds.