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  X  X  X  X  X  X  X  X  X  X  X  X  X  X 2X+2  X  X  X  X  X  X  X  X
 0 2X  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X  0  0  0  0  0 2X 2X  0 2X 2X  0  0  0 2X 2X 2X 2X  0  0 2X  0  0 2X
 0  0 2X  0  0  0  0  0  0  0 2X 2X 2X 2X 2X 2X 2X 2X 2X  0 2X 2X  0  0 2X 2X 2X 2X  0  0  0  0  0  0  0  0 2X 2X  0 2X 2X  0  0 2X 2X 2X  0 2X  0  0  0  0 2X 2X  0
 0  0  0 2X  0  0  0 2X 2X 2X 2X 2X  0 2X 2X  0  0  0  0  0  0  0  0  0 2X 2X 2X 2X 2X 2X 2X 2X  0  0 2X 2X 2X  0 2X  0 2X  0  0 2X  0  0 2X 2X  0 2X 2X 2X 2X 2X 2X
 0  0  0  0 2X  0 2X 2X 2X  0  0  0  0 2X 2X 2X 2X  0  0  0 2X 2X 2X 2X 2X 2X  0  0  0  0 2X 2X  0 2X 2X  0  0  0 2X 2X 2X  0 2X  0  0  0 2X  0 2X 2X  0  0 2X 2X 2X
 0  0  0  0  0 2X 2X  0 2X 2X  0 2X 2X 2X  0  0 2X  0 2X 2X 2X  0  0 2X  0 2X 2X  0  0 2X 2X  0 2X 2X  0  0  0  0  0 2X 2X 2X  0 2X  0 2X  0  0 2X 2X 2X  0 2X  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+50x^52+64x^54+256x^55+118x^56+20x^60+1x^64+2x^76

The gray image is a code over GF(2) with n=440, k=9 and d=208.
This code was found by Heurico 1.16 in 0.094 seconds.