The generator matrix

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

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+47x^50+104x^51+102x^52+224x^53+377x^54+402x^55+363x^56+208x^57+77x^58+52x^59+32x^60+24x^61+19x^62+2x^63+5x^64+8x^65+1x^96

The gray image is a linear code over GF(2) with n=440, k=11 and d=200.
This code was found by Heurico 1.16 in 0.203 seconds.