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

generates a code of length 56 over Z2[X]/(X^4) who´s minimum homogenous weight is 52.

Homogenous weight enumerator: w(x)=1x^0+20x^52+52x^53+66x^54+350x^55+88x^56+324x^57+52x^58+24x^59+25x^60+8x^61+2x^62+10x^63+1x^64+1x^100

The gray image is a linear code over GF(2) with n=448, k=10 and d=208.
This code was found by Heurico 1.16 in 0.125 seconds.