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

generates a code of length 97 over Z2[X]/(X^4) who´s minimum homogenous weight is 96.

Homogenous weight enumerator: w(x)=1x^0+13x^96+222x^97+14x^98+2x^112+2x^113+2x^114

The gray image is a linear code over GF(2) with n=776, k=8 and d=384.
This code was found by Heurico 1.16 in 0.828 seconds.