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 X^2  1
 0  X  0  X  0  0  X  X X^2 X^2+X X^2 X^2+X X^2 X^2 X^2+X X^2+X  0  0  X  2  X X^2 X^2 X^2+X X^2+X  0 X^2  0 X^2 X+2  X X^2+X+2 X^2+X X^2+X+2 X^2+X X+2  X X^2+2  2  2 X^2+2 X^2+2  2 X^2+X+2 X+2 X^2+2 X+2  2 X^2+X+2 X^2+2 X^2+X+2 X+2 X^2+2  2 X+2  X  2
 0  0  X  X X^2+2 X^2+X X^2+X+2 X^2 X^2 X^2+X X+2  2 X^2+X+2  2 X+2 X^2+2  2 X^2+X+2 X+2  X X^2+2 X^2+X X^2+2  0 X^2+X+2 X+2  0 X^2  X X^2+X  2  X  X X^2 X^2 X^2+2 X^2+X X^2+2 X^2+X+2  2 X+2  2  X  2 X^2+X+2 X^2+X  0 X^2 X^2+X X^2+X+2  0  X X^2 X^2+X  0  X X^2+2

generates a code of length 57 over Z4[X]/(X^3+2,2X) 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 9.14 seconds.