The generator matrix

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

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+458x^52+722x^53+1702x^54+890x^55+1531x^56+596x^57+982x^58+368x^59+489x^60+174x^61+171x^62+54x^63+40x^64+12x^65+1x^68+1x^70

The gray image is a linear code over GF(2) with n=448, k=13 and d=208.
This code was found by Heurico 1.16 in 3.61 seconds.