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

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

Homogenous weight enumerator: w(x)=1x^0+245x^96+8x^104+2x^112

The gray image is a linear code over GF(2) with n=768, k=8 and d=384.
This code was found by Heurico 1.16 in 1.06 seconds.