The generator matrix

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

generates a code of length 55 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 52.

Homogenous weight enumerator: w(x)=1x^0+672x^52+584x^53+756x^54+608x^55+544x^56+256x^57+272x^58+128x^59+166x^60+24x^61+68x^62+16x^64+1x^72

The gray image is a code over GF(2) with n=440, k=12 and d=208.
This code was found by Heurico 1.16 in 17 seconds.