The generator matrix

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

generates a code of length 81 over Z2[X]/(X^4) who´s minimum homogenous weight is 74.

Homogenous weight enumerator: w(x)=1x^0+384x^74+1686x^75+3155x^76+4230x^77+5534x^78+7102x^79+7457x^80+7566x^81+7270x^82+6558x^83+5548x^84+3800x^85+2443x^86+1526x^87+634x^88+300x^89+176x^90+100x^91+27x^92+20x^93+9x^94+4x^95+2x^97+2x^100+2x^101

The gray image is a linear code over GF(2) with n=648, k=16 and d=296.
This code was found by Heurico 1.16 in 44.6 seconds.