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

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+409x^76+456x^78+160x^79+833x^80+448x^81+808x^82+160x^83+453x^84+280x^86+54x^88+24x^90+9x^92+1x^148

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 27.2 seconds.