The generator matrix

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

generates a code of length 32 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 27.

Homogenous weight enumerator: w(x)=1x^0+522x^27+2187x^28+4306x^29+8052x^30+10758x^31+13668x^32+10970x^33+8498x^34+4094x^35+1765x^36+526x^37+120x^38+50x^39+11x^40+6x^41+2x^42

The gray image is a code over GF(2) with n=256, k=16 and d=108.
This code was found by Heurico 1.16 in 15.3 seconds.