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  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 X^2  X  1 X^2  X  1 X^2  1
 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 X^3 X^3+X^2 X^3+X^2 X^3+X^2  0 X^3 X^3 X^3+X^2 X^3+X^2 X^2 X^3 X^3  0 X^2 X^3 X^3+X^2  0 X^3 X^3+X^2 X^3+X^2  0  0 X^2 X^3+X^2  0 X^3+X^2 X^3  0 X^2 X^3+X^2 X^2 X^3 X^2 X^2 X^3 X^3  0 X^2  0 X^2 X^3  0 X^3+X^2 X^3 X^3+X^2 X^2 X^3 X^3+X^2  0 X^2 X^3+X^2 X^2 X^2 X^3+X^2 X^3 X^3+X^2 X^3 X^3 X^3 X^2  0  0  0  0 X^3+X^2 X^3 X^3  0 X^2 X^3 X^3+X^2 X^2  0 X^2 X^3 X^3 X^2  0  0  0
 0  0 X^3+X^2  0 X^2 X^2 X^3+X^2  0 X^2  0  0 X^3+X^2 X^2 X^3+X^2  0  0  0 X^3 X^3+X^2 X^3 X^3 X^2 X^2 X^2 X^3+X^2 X^3  0 X^2  0 X^3+X^2 X^2 X^3 X^2 X^3  0 X^2 X^3+X^2 X^3 X^3 X^3+X^2 X^3+X^2  0  0 X^2 X^2 X^3 X^2  0 X^3+X^2 X^3+X^2 X^3 X^3 X^3+X^2  0 X^3  0 X^3+X^2 X^2 X^3 X^3+X^2 X^3+X^2 X^3+X^2  0 X^3 X^2  0 X^2 X^2 X^2  0 X^3 X^2 X^3  0 X^3+X^2  0 X^3  0 X^3 X^3+X^2  0 X^3+X^2 X^3+X^2 X^3+X^2  0 X^3 X^3+X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^3 X^2 X^2 X^3 X^3 X^3
 0  0  0 X^3+X^2 X^2  0 X^3+X^2 X^2 X^2  0 X^3+X^2  0  0 X^3+X^2 X^2 X^3  0 X^3  0 X^3+X^2 X^3+X^2 X^3+X^2  0 X^2 X^3 X^2 X^3+X^2 X^3+X^2 X^3 X^3+X^2  0 X^3 X^3+X^2 X^2 X^3+X^2 X^3+X^2 X^3  0  0 X^3 X^3+X^2 X^2 X^3  0 X^3  0 X^2 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^2 X^2  0  0 X^2 X^3 X^3+X^2 X^2 X^2 X^3 X^2 X^3 X^3 X^2 X^3  0 X^3 X^2 X^3 X^3 X^3+X^2  0 X^2 X^3+X^2 X^2  0  0 X^2 X^2 X^3 X^3 X^3 X^3 X^3 X^3 X^2  0
 0  0  0  0 X^3  0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0 X^3  0  0 X^3 X^3 X^3  0 X^3 X^3  0  0  0 X^3 X^3  0  0 X^3 X^3  0  0  0  0 X^3 X^3  0 X^3 X^3  0 X^3 X^3  0  0 X^3  0 X^3 X^3  0 X^3 X^3  0  0  0 X^3  0 X^3  0 X^3  0 X^3  0  0 X^3 X^3 X^3  0 X^3  0  0 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3  0 X^3 X^3 X^3  0  0 X^3  0  0  0 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+16x^90+44x^91+86x^92+120x^93+154x^94+348x^95+516x^96+352x^97+160x^98+116x^99+64x^100+40x^101+21x^102+4x^103+2x^104+2x^108+1x^112+1x^182

The gray image is a linear code over GF(2) with n=768, k=11 and d=360.
This code was found by Heurico 1.16 in 1.23 seconds.