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

generates a code of length 90 over Z2[X]/(X^3) who´s minimum homogenous weight is 89.

Homogenous weight enumerator: w(x)=1x^0+72x^89+11x^90+16x^91+12x^92+4x^94+3x^96+8x^97+1x^122

The gray image is a linear code over GF(2) with n=360, k=7 and d=178.
This code was found by Heurico 1.16 in 1.03 seconds.