The generator matrix

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

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+18x^52+104x^53+45x^54+688x^55+45x^56+104x^57+18x^58+1x^110

The gray image is a code over GF(2) with n=440, k=10 and d=208.
This code was found by Heurico 1.16 in 0.094 seconds.