The generator matrix

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

generates a code of length 55 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 51.

Homogenous weight enumerator: w(x)=1x^0+80x^51+544x^52+604x^53+875x^54+546x^55+480x^56+264x^57+272x^58+126x^59+177x^60+60x^61+43x^62+12x^63+6x^64+4x^65+1x^66+1x^70

The gray image is a code over GF(2) with n=440, k=12 and d=204.
This code was found by Heurico 1.16 in 0.172 seconds.