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

generates a code of length 73 over Z2[X]/(X^4) who´s minimum homogenous weight is 69.

Homogenous weight enumerator: w(x)=1x^0+160x^69+91x^70+374x^71+308x^72+310x^73+291x^74+276x^75+40x^76+84x^77+33x^78+54x^79+2x^80+22x^81+1x^82+1x^128

The gray image is a linear code over GF(2) with n=584, k=11 and d=276.
This code was found by Heurico 1.16 in 0.453 seconds.