The generator matrix

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

generates a code of length 55 over Z2[X]/(X^4) who´s minimum homogenous weight is 50.

Homogenous weight enumerator: w(x)=1x^0+116x^50+284x^51+323x^52+504x^53+593x^54+676x^55+518x^56+464x^57+193x^58+156x^59+99x^60+72x^61+56x^62+20x^63+19x^64+1x^66+1x^86

The gray image is a linear code over GF(2) with n=440, k=12 and d=200.
This code was found by Heurico 1.16 in 0.375 seconds.