The generator matrix

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

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+27x^50+48x^51+59x^52+78x^53+112x^54+408x^55+654x^56+394x^57+102x^58+42x^59+24x^60+28x^61+26x^62+12x^63+9x^64+10x^65+2x^66+2x^67+5x^68+2x^69+2x^70+1x^98

The gray image is a linear code over GF(2) with n=448, k=11 and d=200.
This code was found by Heurico 1.16 in 0.219 seconds.