The generator matrix

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

generates a code of length 56 over Z2[X]/(X^4) who´s minimum homogenous weight is 50.

Homogenous weight enumerator: w(x)=1x^0+708x^50+1944x^51+3665x^52+5826x^53+7000x^54+8936x^55+9184x^56+9516x^57+7448x^58+5450x^59+3098x^60+1474x^61+776x^62+324x^63+97x^64+48x^65+20x^66+18x^67+1x^68+2x^72

The gray image is a linear code over GF(2) with n=448, k=16 and d=200.
This code was found by Heurico 1.16 in 30.8 seconds.