The generator matrix

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

generates a code of length 55 over Z4[X]/(X^2+2) 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 0.125 seconds.