The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  1  1  X  1  1  X  1  1  X  1  X  1  X  X 2X+2  1  X  X 2X+2  X  1  X  X 2X  0 2X+2 2X+2  X 2X+2  X  X 2X  1 2X+2 2X  0
 0 2X  0 2X  0 2X  0 2X  0 2X  0 2X  0 2X  0 2X  0 2X  0 2X 2X  0 2X 2X  0 2X 2X  0 2X 2X  0  0  0 2X 2X 2X  0 2X 2X  0  0 2X  0 2X 2X  0 2X  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 57.

Homogenous weight enumerator: w(x)=1x^0+20x^57+2x^58+4x^59+5x^60

The gray image is a code over GF(2) with n=448, k=5 and d=228.
This code was found by Heurico 1.16 in 0.078 seconds.