The generator matrix

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

generates a code of length 55 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 52.

Homogenous weight enumerator: w(x)=1x^0+640x^52+684x^53+720x^54+588x^55+485x^56+252x^57+288x^58+180x^59+156x^60+24x^61+72x^62+4x^64+2x^68

The gray image is a code over GF(2) with n=440, k=12 and d=208.
This code was found by Heurico 1.16 in 5.02 seconds.