The generator matrix

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

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+61x^52+120x^54+640x^55+155x^56+40x^58+6x^60+1x^108

The gray image is a code over GF(2) with n=440, k=10 and d=208.
This code was found by Heurico 1.16 in 3.44 seconds.