The generator matrix

 1  0  0  1  1  1  2  2 2X+2  1  1  2  1  1 3X  1  1 3X  X  1  1  1 X+2  1 3X+2  1 3X+2 2X 2X X+2  1  1  1  1 2X+2  X 3X  1  1  1  1 3X+2 2X+2  1  1  1 2X 3X+2 2X  1  1  0  0  1 3X  1
 0  1  0  0  3  3  1  X  1 2X 2X+3  1  2  1 3X+2 3X 3X+3  1  1 3X+3 3X+2 3X+1  1 X+1 2X+2 X+2  1  1  1  X  1 2X X+3 2X+1 2X+2  1  1 2X+2 X+3 3X+1  X  0 X+2 3X 2X+2 3X+1  1  1  1 2X+2 2X+3  1  1 3X  1 2X+3
 0  0  1 X+1 3X+1 2X 3X+3  1 3X  X 3X  3  3 2X+3  1  1  2  3 3X 2X+1  X 3X+1  0  2  1 X+1 3X+3  2 3X+1  1  0  3 X+1 X+3  1  0 2X+1  X 2X+3 3X 3X  1  1  3 2X X+2 3X+2 X+2 2X+3 X+1 3X X+3 2X+1  1 X+1 X+3
 0  0  0 2X 2X  0 2X 2X 2X 2X 2X  0  0  0 2X 2X 2X 2X  0  0 2X 2X 2X  0  0  0 2X 2X  0  0 2X 2X  0  0  0  0  0  0 2X 2X  0 2X  0  0  0  0  0  0 2X 2X  0 2X 2X  0 2X  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+569x^52+624x^53+1592x^54+912x^55+1534x^56+696x^57+968x^58+440x^59+436x^60+120x^61+192x^62+24x^63+81x^64+3x^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 2.81 seconds.