The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+100x^54+64x^55+74x^56+12x^58+5x^64

The gray image is a code over GF(2) with n=440, k=8 and d=216.
This code was found by Heurico 1.16 in 0.125 seconds.