The generator matrix

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

generates a code of length 55 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 53.

Homogenous weight enumerator: w(x)=1x^0+340x^53+126x^54+112x^55+124x^56+292x^57+2x^58+24x^59+1x^64+2x^76

The gray image is a code over GF(2) with n=440, k=10 and d=212.
This code was found by Heurico 1.16 in 0.125 seconds.