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

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

Homogenous weight enumerator: w(x)=1x^0+229x^52+32x^53+224x^54+352x^55+383x^56+352x^57+224x^58+32x^59+208x^60+7x^64+3x^68+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 0.25 seconds.