The generator matrix

 1  0  1  1  1 X+2  1  X  1 2X  1  1 2X+2  1  1  1 3X+2  1  1  2  1 3X  1  1  1  1  1  1  1  1  2  1  1 3X  1  1 3X+2  1  1  0  1  1  1  X  X 2X 2X+2  1  1  1  1  1  1  1  1  1
 0  1 X+1 3X+2  3  1  2  1 3X+3  1 3X  1  1 2X X+1 X+2  1 X+3 2X+2  1  X  1 X+1 X+3 2X+3 2X+1 2X+3 2X+1 3X+1 2X  1 X+3 X+2  1  3 2X+2  1 2X+3  X  1 X+1  1 2X+1 3X+2 3X  1  1 3X+1 3X+2 3X 3X+3 3X+1  3  3 X+1 X+1
 0  0 2X+2  0  2 2X+2  0 2X+2  2  2  0 2X+2  2 2X+2 2X  2  0 2X 2X+2  0  2  0 2X 2X  0 2X 2X  0 2X+2 2X+2  0  2  2  0 2X+2  2 2X  2 2X+2 2X 2X 2X  0 2X  2  2 2X+2  2  0 2X 2X  0  2  2 2X+2  2
 0  0  0 2X  0  0  0  0 2X 2X 2X 2X 2X  0 2X 2X 2X  0 2X 2X  0  0 2X  0  0 2X 2X  0  0  0 2X 2X 2X  0  0 2X 2X 2X  0  0  0  0 2X 2X  0  0  0 2X  0  0 2X 2X 2X  0 2X 2X
 0  0  0  0 2X  0 2X 2X  0  0 2X 2X 2X 2X  0 2X  0  0  0 2X  0 2X 2X 2X  0 2X  0 2X 2X  0  0 2X  0  0  0 2X 2X  0 2X 2X 2X  0 2X  0 2X 2X  0  0  0 2X  0  0 2X  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+389x^52+240x^53+538x^54+576x^55+763x^56+480x^57+550x^58+192x^59+237x^60+48x^61+54x^62+14x^64+2x^66+2x^68+8x^70+2x^80

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 50.8 seconds.