The generator matrix

 1  0  0  1  1  1 2X  1  1  0  1  1  2 X+2  1 3X+2 3X  X  1  1  1  1  1 2X  1  1  1  1  2 X+2  1  1  1  X 2X+2  1  1 2X+2  1  1  1  1  1 3X  2  1  X  1  1  1  X  1  X 3X 2X+2  1
 0  1  0 2X 2X+3  3  1  X 3X 3X 3X+3 X+3  1  1 2X+2  1 3X+2  1  1 3X+2  X  3 3X+1  1 3X+3 2X+1  2 3X  1  1 2X+2 3X+1 X+2 2X+2  1 3X+2  3  1 3X+1 3X+2 2X  1 X+1  1  0 X+3  1 2X+1  0  2  X 3X+1  1  1 3X  3
 0  0  1 3X+1 X+1 2X 3X+1 3X 2X+3  1  3  X X+2 2X+1 3X X+2  1 X+1 3X+2 3X+1  2 2X+1 X+2 2X+1 X+1 2X+2  1  0  0  2  2 2X 2X+1  1 X+1 2X+3 X+2 3X  3 3X+3  X X+3 2X+2  X  1 X+2 3X+3  0 2X+1 X+1  1 X+1  3 2X  1 3X

generates a code of length 56 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 53.

Homogenous weight enumerator: w(x)=1x^0+644x^53+650x^54+792x^55+504x^56+548x^57+263x^58+296x^59+141x^60+124x^61+37x^62+80x^63+1x^64+12x^65+1x^66+1x^68+1x^70

The gray image is a code over GF(2) with n=448, k=12 and d=212.
This code was found by Heurico 1.16 in 31.1 seconds.