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

generates a code of length 98 over Z2[X]/(X^4) who´s minimum homogenous weight is 94.

Homogenous weight enumerator: w(x)=1x^0+96x^94+32x^95+208x^96+96x^97+220x^98+96x^99+136x^100+32x^101+56x^102+34x^104+12x^106+2x^112+2x^120+1x^128

The gray image is a linear code over GF(2) with n=784, k=10 and d=376.
This code was found by Heurico 1.16 in 0.922 seconds.