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

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+256x^52+32x^53+80x^54+352x^55+613x^56+352x^57+80x^58+32x^59+240x^60+9x^64+1x^104

The gray image is a code over GF(2) with n=448, k=11 and d=208.
This code was found by Heurico 1.16 in 0.437 seconds.