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

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+70x^89+140x^90+16x^91+12x^92+4x^94+3x^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 2 seconds.