The generator matrix

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

generates a code of length 81 over Z2[X]/(X^4) who´s minimum homogenous weight is 77.

Homogenous weight enumerator: w(x)=1x^0+184x^77+585x^78+762x^79+656x^80+440x^81+365x^82+330x^83+268x^84+168x^85+88x^86+80x^87+104x^88+44x^89+8x^90+8x^93+1x^94+2x^96+1x^100+1x^102

The gray image is a linear code over GF(2) with n=648, k=12 and d=308.
This code was found by Heurico 1.16 in 1.63 seconds.