The generator matrix

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

generates a code of length 57 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 55.

Homogenous weight enumerator: w(x)=1x^0+132x^55+209x^56+404x^57+152x^58+76x^59+21x^60+28x^61+1x^108

The gray image is a code over GF(2) with n=456, k=10 and d=220.
This code was found by Heurico 1.16 in 9.03 seconds.