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

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+80x^90+222x^91+231x^92+354x^93+431x^94+496x^95+621x^96+488x^97+383x^98+268x^99+201x^100+126x^101+59x^102+60x^103+32x^104+24x^105+1x^106+10x^107+2x^108+4x^110+1x^114+1x^170

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 1.52 seconds.