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  1  1  1  1  X  1
 0  X  2 3X+2  0 3X+2  2 3X  0 3X+2 3X  2 2X X+2 2X+2 3X  0 3X+2  2 3X  0 3X+2  2 3X  0 3X+2  2 3X 2X X+2 2X+2  X  0 3X+2  2 3X 2X X+2 2X+2  X 2X X+2 2X+2  X 2X X+2 2X X+2  0 3X+2  2 2X+2 3X  X 3X+2  0
 0  0 2X  0  0  0 2X  0  0 2X 2X 2X  0 2X 2X 2X 2X  0  0 2X 2X 2X  0  0 2X  0  0 2X 2X 2X  0  0  0 2X 2X  0 2X 2X 2X 2X  0  0 2X  0  0 2X  0  0 2X 2X  0  0  0 2X  0  0
 0  0  0 2X  0  0  0 2X 2X 2X 2X 2X 2X  0 2X  0 2X  0  0 2X 2X 2X  0  0  0 2X 2X  0  0  0 2X 2X 2X  0 2X 2X 2X 2X  0  0 2X  0  0 2X  0 2X  0  0  0  0 2X 2X  0 2X  0  0
 0  0  0  0 2X 2X 2X 2X 2X  0 2X  0  0 2X 2X  0  0  0  0  0 2X 2X 2X 2X 2X 2X 2X 2X  0  0  0  0  0 2X 2X  0  0 2X  0  0 2X  0 2X 2X 2X  0  0 2X  0  0  0 2X  0 2X 2X  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+7x^52+32x^53+34x^54+192x^55+495x^56+192x^57+28x^58+32x^59+9x^60+1x^62+1x^110

The gray image is a code over GF(2) with n=448, k=10 and d=208.
This code was found by Heurico 1.16 in 0.156 seconds.