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

generates a code of length 82 over Z2[X]/(X^4) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+180x^78+64x^79+437x^80+128x^81+456x^82+320x^83+224x^84+132x^86+105x^88+1x^152

The gray image is a linear code over GF(2) with n=656, k=11 and d=312.
This code was found by Heurico 1.16 in 137 seconds.