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

generates a code of length 56 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 52.

Homogenous weight enumerator: w(x)=1x^0+235x^52+24x^53+328x^54+248x^55+507x^56+200x^57+262x^58+40x^59+115x^60+44x^62+35x^64+6x^66+2x^68+1x^96

The gray image is a code over GF(2) with n=448, k=11 and d=208.
This code was found by Heurico 1.16 in 3.14 seconds.