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

generates a code of length 90 over Z4[X]/(X^3+2,2X) 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 code over GF(2) with n=720, k=8 and d=356.
This code was found by Heurico 1.16 in 98.3 seconds.