The generator matrix

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

generates a code of length 56 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 50.

Homogenous weight enumerator: w(x)=1x^0+630x^50+2064x^51+3752x^52+5220x^53+7694x^54+8868x^55+9506x^56+8868x^57+7692x^58+5080x^59+3153x^60+1780x^61+826x^62+204x^63+135x^64+36x^65+14x^66+8x^67+5x^68

The gray image is a code over GF(2) with n=448, k=16 and d=200.
This code was found by Heurico 1.16 in 29.5 seconds.