The generator matrix

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

generates a code of length 81 over Z2[X]/(X^4) who´s minimum homogenous weight is 72.

Homogenous weight enumerator: w(x)=1x^0+320x^72+1448x^73+3472x^74+6204x^75+10134x^76+14774x^77+20821x^78+25644x^79+30561x^80+32506x^81+33083x^82+27114x^83+21193x^84+15086x^85+9558x^86+5098x^87+2593x^88+1360x^89+708x^90+252x^91+105x^92+40x^93+33x^94+6x^95+21x^96+2x^97+5x^98+2x^99

The gray image is a linear code over GF(2) with n=648, k=18 and d=288.
This code was found by Heurico 1.16 in 719 seconds.