The generator matrix

 1  0  0  1  1  1  2  2 2X+2  1  1  2  1  1 3X  1  1 X+2  1  1  X  1 3X+2  1  2  1  1  1  1  1 3X+2  1  1  X  1  0  2  2  1 2X+2 3X+2  1  1  1 3X+2  1  1 3X+2  1  1  1  X  1  1  0  1
 0  1  0  0 2X+3 2X+3  1 3X  1 2X  3  1  2 2X+1 3X+2  X 3X+1  1 3X X+1  1 3X+2  1  3  1 X+3 3X 3X+3  X 2X+1 2X+2 3X+1 2X+2  1  3  1 2X+2  1 X+3  1 2X 3X+1 3X+3 X+2  1 2X  1  1  X 2X  0  1 3X+1 3X+2  1  3
 0  0  1 X+1 3X+1 2X X+3  1  X 3X  X  3 2X+3  3  1 2X+1 3X X+3  2  3  0 X+2  X 3X+2 3X+1 3X+3 2X+3  X X+3  1  1 2X+2 2X 2X+3  0  3  1  2 X+1 X+2  1  0 2X+3 3X+3  2 3X+1 2X 3X X+2  3 3X 3X+1 3X+2 3X X+3 X+3
 0  0  0 2X 2X  0 2X 2X 2X 2X 2X  0  0  0  0 2X  0  0 2X 2X  0  0 2X  0  0 2X  0 2X 2X  0 2X  0 2X 2X 2X 2X 2X 2X  0  0  0 2X  0  0 2X  0  0  0  0 2X  0  0 2X 2X 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+604x^52+704x^53+1428x^54+968x^55+1418x^56+784x^57+936x^58+416x^59+542x^60+176x^61+132x^62+24x^63+53x^64+6x^68

The gray image is a code over GF(2) with n=448, k=13 and d=208.
This code was found by Heurico 1.16 in 1.11 seconds.