The generator matrix

 1  0  1  1  1 X^2+X  1  1 X^3+X^2  1  1 X^3+X  1  1  0  1  1 X^2+X  1  1 X^3+X^2  1  1 X^3+X  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  0 X^2+X X^3+X^2 X^2 X^3+X^2+X X^3 X^3+X  X  1  1  1  1  1  1  1
 0  1 X+1 X^2+X X^2+1  1 X^3+X^2 X^3+X^2+X+1  1 X^3+X X^3+1  1  0 X+1  1 X^2+X X^2+1  1 X^3+X^2 X^3+X^2+X+1  1 X^3+X X^3+1  1  0 X^2+X X+1 X^3+1 X^3+X^2 X^3+X^2+X X^3+X^2+X+1 X^2+1 X^3+X X^3 X^2  X X^3+X+1 X^3+X^2+1 X^2+X+1  1  1  1  1  1  1  1  1  1  0 X^3 X^3+X^2+X X^2+X X^3 X^2  0
 0  0 X^3  0 X^3  0 X^3  0 X^3 X^3  0 X^3  0  0  0 X^3  0  0 X^3 X^3 X^3  0 X^3 X^3 X^3  0  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3 X^3  0 X^3  0 X^3 X^3  0  0 X^3 X^3  0  0 X^3 X^3  0 X^3  0 X^3  0
 0  0  0 X^3 X^3 X^3 X^3  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0 X^3 X^3  0  0  0 X^3  0  0 X^3 X^3  0 X^3  0 X^3  0  0 X^3 X^3 X^3  0  0 X^3  0  0 X^3 X^3  0  0 X^3  0 X^3 X^3  0  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+24x^52+328x^53+38x^54+240x^55+38x^56+328x^57+24x^58+1x^64+2x^78

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 0.093 seconds.