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

generates a code of length 96 over Z2[X]/(X^4) who´s minimum homogenous weight is 90.

Homogenous weight enumerator: w(x)=1x^0+102x^90+220x^91+280x^92+348x^93+490x^94+392x^95+511x^96+516x^97+452x^98+224x^99+200x^100+140x^101+78x^102+56x^103+29x^104+12x^105+22x^106+4x^107+10x^108+8x^109+1x^160

The gray image is a linear code over GF(2) with n=768, k=12 and d=360.
This code was found by Heurico 1.16 in 1.5 seconds.