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

generates a code of length 88 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 85.

Homogenous weight enumerator: w(x)=1x^0+80x^85+178x^86+230x^87+206x^88+88x^89+108x^90+46x^91+46x^92+24x^93+1x^94+8x^95+2x^96+4x^97+1x^98+1x^132

The gray image is a code over GF(2) with n=704, k=10 and d=340.
This code was found by Heurico 1.16 in 0.5 seconds.