The generator matrix

 1  0  0  1  1  1  0 X^3  1  1 X^3 X^2  1  1  1  1 X^3+X X^3+X  1 X^3+X  1  1  1 X^3+X^2+X X^2+X  1 X^3+X^2  1  1  1 X^3+X^2+X  1  1  1 X^3+X X^3+X  1  X X^2  1  1  1 X^2  1 X^3  1  X X^3  1  1  1  X  1 X^3+X^2 X^2+X X^2 X^3 X^3+X  1  1 X^2+X X^3+X^2+X X^3+X^2+X X^2+X  1 X^2  0  1  1  1  1 X^3+X  1  1  1  1  1  1  1 X^3+X  1
 0  1  0  0 X^2+1 X^2+1  1 X^3+X^2+X X^3 X^3+X^2+1  1  1 X^3+X^2  1 X^2+X X+1  1 X^2+X  X  1 X^3+X+1 X^3+X^2+X+1 X^2+X+1  0  1 X^3+X^2  1 X^3 X^3+X^2 X^3+X+1  1 X^3+X^2+X X^3+X  X X^3+X  1 X^2+X  1  1  1 X^3+X^2+X+1 X^2+1  0 X^3+X^2+1  1 X^3+1  1 X^3+X X^2+X  X X^3+X^2+X+1  1 X^3+X+1  1 X^3+X  1  1  1 X^2  0  1  1 X^3 X^2 X^3 X^3+X^2+X  1 X^2+X X^3+X+1  1 X^2  1 X+1 X^3+X^2+X X^3+1 X^2 X^2+1 X^3+X^2 X^3+X  1 X^2
 0  0  1 X+1 X^3+X+1 X^3 X^3+X^2+X+1  1 X^3+X^2+X X^2+1  1 X^3+X X^3+X^2+1  X X^3+X+1 X^2 X^3+X^2+1  1  X X^3+X^2+X X^2+X+1 X^3+X^2+X X^2+1  1  0 X+1 X^2+1 X^3+X^2  1 X^3+X X^2+X+1 X^3+X^2 X^3+X^2 X^3+X^2+X+1  1 X^2 X^3+X^2+1  1  0 X^3+X^2+X X^3+X^2+X+1 X^3+X^2+1  1 X^2 X^3+X^2+1 X^2+X+1 X^3+X^2+X  1 X^3+X^2+X X^3+X^2+1 X^3 X^3+X^2+X+1  1 X^2+1  1  X X^3+X^2+X+1 X^3+X+1 X^2+X+1  X X^2 X^3+X+1  1  1 X^3+1  1 X^3+X+1 X+1 X^3  0 X^2 X^2+1 X^2 X^2+X+1 X^3+1 X^3+X^2+X X^3+X^2+X+1 X+1 X^3+X^2  1 X^2+X
 0  0  0 X^3 X^3  0 X^3 X^3 X^3  0  0 X^3  0 X^3  0 X^3 X^3  0  0  0  0  0 X^3  0 X^3  0 X^3 X^3 X^3 X^3  0  0 X^3 X^3 X^3  0  0  0 X^3  0 X^3 X^3 X^3 X^3 X^3  0 X^3  0 X^3 X^3  0 X^3  0  0  0  0  0 X^3  0  0  0  0 X^3  0  0 X^3 X^3 X^3 X^3  0 X^3  0  0 X^3  0  0 X^3 X^3  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+144x^76+762x^77+890x^78+1264x^79+882x^80+1172x^81+722x^82+780x^83+427x^84+430x^85+252x^86+196x^87+101x^88+100x^89+25x^90+32x^91+5x^92+4x^94+3x^98

The gray image is a linear code over GF(2) with n=648, k=13 and d=304.
This code was found by Heurico 1.16 in 14.2 seconds.