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

generates a code of length 55 over Z2[X]/(X^4) who´s minimum homogenous weight is 52.

Homogenous weight enumerator: w(x)=1x^0+40x^52+56x^53+184x^54+464x^55+185x^56+56x^57+32x^58+5x^60+1x^108

The gray image is a linear code over GF(2) with n=440, k=10 and d=208.
This code was found by Heurico 1.16 in 0.187 seconds.