The generator matrix

 1  0  0  1  1  1 2X  1  1  1  1 2X+2  2  1  1 X+2  1 3X+2  X  1  1 2X 3X+2 3X+2  1  1  0  1  1  0 X+2 3X+2 2X+2  1  1  1  2 2X  1  1 X+2  1 3X+2  1 3X  1  1  1  1  1  2  X  X  0  2  1
 0  1  0 2X 2X+3  3  1  X 3X+3 3X X+3  1  X 3X+1  0  1 2X+1  X  1 X+2  X  1  1  2 3X+3  3  1 X+2 2X X+2  1  1 2X+2 3X  1 3X+1  1  1 X+1  1 X+2 X+3  1 2X+1  1  3 2X+2 X+2  2 3X+1  1  1 X+2  1  0  0
 0  0  1 3X+1 X+1 2X 3X+1 3X  1 2X+1  X  X  1 2X+2 3X+2 X+2 2X+3  1 2X+1 3X+1  2 2X+3 X+3  1 X+1 3X 2X+2  3 2X+3  1 2X+2  1  1 X+1 2X 2X+3  1 3X+3 X+3 X+2  1 X+2 X+1  1  2 2X+2 2X+1 2X+1 3X+3 X+1  0 3X+1  1 3X  1  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+72x^52+556x^53+704x^54+728x^55+554x^56+484x^57+324x^58+218x^59+172x^60+156x^61+42x^62+58x^63+16x^64+8x^65+1x^66+1x^68+1x^70

The gray image is a code over GF(2) with n=448, k=12 and d=208.
This code was found by Heurico 1.16 in 0.187 seconds.