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

generates a code of length 56 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 52.

Homogenous weight enumerator: w(x)=1x^0+36x^52+950x^56+36x^60+1x^112

The gray image is a code over GF(2) with n=448, k=10 and d=208.
This code was found by Heurico 1.16 in 0.11 seconds.