The generator matrix

 1  0  0  1  1  1  X  1  1 X^2+X  1 X^3+X X^3  1  0  1  1 X^3+X^2  1  1  1 X^2+X X^2+X  1  1  1 X^2+X  1 X^3+X^2+X X^3+X X^3 X^3 X^3+X^2  1 X^3  1  1  1  1  1  X  1  1 X^2 X^2  1 X^3+X  1  1 X^3+X^2 X^2+X  1  1  1  1  1
 0  1  0  0 X^2+1 X^3+X+1  1 X^3+X^2+X+1 X^2+X X^3+X^2 X^3+X^2+X+1  1  1 X^2+X X^3  1 X^3  1 X^3+X X^3+X^2+X+1 X^2+1  1  1 X^2+X X^2 X^3+X^2+1 X^2+X X^3+X^2+1  1  1  1  1 X^3+X^2+X X^2+X+1 X^2 X^3+X^2+X+1 X^3+1 X^2 X^2+X+1  X X^2  1 X^3+X^2+X  1  1 X^2+X+1  1  0 X^3+X+1 X^3+X  1 X^3 X^3+X^2+X  0  X X^3
 0  0  1  1  1 X^2 X^2+1 X^3+X+1 X^3+1  1 X^3+X^2+X X+1 X^3+X^2+X X^2+X  1 X^2+X X^2+X+1 X^3+X^2+X+1 X^3 X^3+1 X^3+X+1 X^3 X^3+X^2+1 X^3+X X^3+X+1 X^3+X^2+X  1 X^3+X^2+1 X^3+X^2 X^2+X+1 X^2+1 X^3+X^2+X  1 X^3+1  1 X^3 X+1 X^3+X+1 X^2+X X^3+1  1 X^3 X^3+X^2 X^3 X^3+1 X^3+X  1 X^2+X+1 X+1  1 X^3+X+1 X^3  0 X^3+X^2 X^3+X+1 X^3
 0  0  0  X X^3+X X^3 X^3+X X^3+X^2+X  X  X X^3+X^2 X^3+X^2+X  0 X^3 X^2+X X^2 X^2+X X^3+X X^2 X^2+X X^3+X^2+X  0 X^3+X^2 X^2+X X^2 X^2+X X^2  0 X^3+X^2+X  0 X^3 X^2+X X^2  0 X^2 X^3+X X^2  X X^2+X X^3 X^3 X^2+X X^3+X X^3+X^2  X X^3 X^2+X X^3 X^3 X^2+X X^3+X^2 X^3+X^2 X^3+X^2+X  0 X^2  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+160x^50+1108x^51+1701x^52+3130x^53+3433x^54+4862x^55+4348x^56+4948x^57+3282x^58+2848x^59+1442x^60+930x^61+267x^62+174x^63+63x^64+48x^65+16x^66+5x^68+2x^70

The gray image is a linear code over GF(2) with n=448, k=15 and d=200.
This code was found by Heurico 1.16 in 10.3 seconds.