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

generates a code of length 98 over Z2[X]/(X^4) who´s minimum homogenous weight is 96.

Homogenous weight enumerator: w(x)=1x^0+286x^96+128x^97+256x^98+128x^99+192x^100+31x^104+1x^112+1x^184

The gray image is a linear code over GF(2) with n=784, k=10 and d=384.
This code was found by Heurico 1.16 in 12.7 seconds.