The generator matrix

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

generates a code of length 73 over Z2[X]/(X^4) who´s minimum homogenous weight is 66.

Homogenous weight enumerator: w(x)=1x^0+226x^66+1454x^67+2808x^68+4234x^69+5176x^70+7452x^71+7194x^72+8908x^73+7443x^74+7344x^75+5080x^76+3726x^77+2073x^78+1364x^79+597x^80+288x^81+91x^82+34x^83+16x^84+12x^85+15x^86

The gray image is a linear code over GF(2) with n=584, k=16 and d=264.
This code was found by Heurico 1.16 in 40.9 seconds.