The generator matrix

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

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+520x^51+764x^52+1272x^53+1248x^54+1366x^55+864x^56+892x^57+470x^58+300x^59+177x^60+216x^61+57x^62+38x^63+1x^64+4x^65+1x^68+1x^70

The gray image is a code over GF(2) with n=440, k=13 and d=204.
This code was found by Heurico 1.16 in 4.77 seconds.