The generator matrix

 1  0  1  1  1 3X+2  1  1 2X  1  1 X+2  1  2  1  1  X  1  1 2X+2  1  1  1 3X  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  0 3X+2  1  1  1  1
 0  1 X+1 X+2  3  1  X X+1  1  2 2X+3  1 3X+3  1 2X 2X+1  1 3X+2 2X+2  1 3X+1 2X+1 3X  1 X+3  1 X+1  3 3X+3 2X+1 2X+3 X+1  0 3X+2 3X+2  2  2 3X 3X+2 3X  0 2X+3  X  0 3X+1 2X+2 2X+2 2X+2 3X+2  1  X  1 3X+1 X+3 3X  0
 0  0 2X+2 2X+2  2  0  2 2X 2X+2  0 2X 2X+2 2X+2  0  2  0  2  0 2X+2  2  2 2X  0  0 2X 2X+2 2X  2  2 2X+2 2X  0  0 2X  2 2X+2 2X 2X+2 2X+2  0 2X+2 2X+2  2 2X  0  2  2  0 2X  2 2X 2X  2  0  2  0
 0  0  0 2X 2X 2X  0 2X  0 2X  0 2X 2X 2X 2X 2X  0 2X  0 2X  0  0  0  0 2X 2X  0  0 2X  0 2X  0 2X  0  0 2X  0 2X  0 2X  0 2X 2X 2X 2X  0 2X  0 2X 2X 2X  0 2X  0 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+83x^52+244x^53+283x^54+376x^55+113x^56+352x^57+255x^58+238x^59+91x^60+4x^61+2x^62+2x^66+2x^67+1x^70+1x^90

The gray image is a code over GF(2) with n=448, k=11 and d=208.
This code was found by Heurico 1.16 in 0.172 seconds.