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

generates a code of length 90 over Z2[X]/(X^4) who´s minimum homogenous weight is 89.

Homogenous weight enumerator: w(x)=1x^0+74x^89+138x^90+8x^91+14x^92+4x^93+6x^94+1x^96+8x^97+2x^105

The gray image is a linear code over GF(2) with n=720, k=8 and d=356.
This code was found by Heurico 1.16 in 96.8 seconds.