The generator matrix

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

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+510x^52+512x^54+581x^56+384x^58+54x^60+1x^64+4x^68+1x^88

The gray image is a linear code over GF(2) with n=440, k=11 and d=208.
This code was found by Heurico 1.16 in 41.4 seconds.