The generator matrix

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

generates a code of length 55 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 54.

Homogenous weight enumerator: w(x)=1x^0+232x^54+64x^55+196x^56+16x^58+2x^68+1x^72

The gray image is a code over GF(2) with n=440, k=9 and d=216.
This code was found by Heurico 1.16 in 0.109 seconds.