The generator matrix

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

generates a code of length 81 over Z2[X]/(X^4) who´s minimum homogenous weight is 76.

Homogenous weight enumerator: w(x)=1x^0+158x^76+332x^77+501x^78+480x^79+469x^80+382x^81+469x^82+396x^83+382x^84+268x^85+152x^86+44x^87+23x^88+10x^89+11x^90+8x^91+2x^92+2x^98+4x^104+1x^110+1x^112

The gray image is a linear code over GF(2) with n=648, k=12 and d=304.
This code was found by Heurico 1.16 in 0.672 seconds.