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  X  1  1  1  X  X  1 X^3  X  1 X^2  1  1  X
 0 X^3+X^2  0  0  0 X^2 X^3+X^2 X^2  0 X^3 X^3+X^2 X^3+X^2  0 X^3 X^3+X^2 X^3+X^2  0 X^3 X^3+X^2 X^2 X^3 X^3 X^3+X^2 X^2 X^3  0  0 X^2 X^3+X^2 X^2  0 X^2 X^3 X^3+X^2  0 X^2 X^3+X^2 X^3+X^2 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0 X^3+X^2 X^3+X^2 X^2 X^2 X^3 X^3 X^3+X^2 X^3 X^3+X^2
 0  0 X^3+X^2  0 X^2 X^2 X^2 X^3  0 X^3 X^2 X^3+X^2 X^2 X^2 X^3 X^3  0 X^3+X^2  0 X^2  0 X^3+X^2 X^2 X^3 X^3+X^2 X^3 X^2  0 X^3+X^2  0  0 X^3+X^2 X^3 X^2 X^3 X^2 X^3+X^2 X^2  0  0  0 X^3 X^3+X^2 X^3+X^2 X^2  0 X^3 X^3  0  0 X^3+X^2 X^2 X^3 X^3 X^3+X^2
 0  0  0 X^3+X^2 X^2 X^3 X^3+X^2 X^3+X^2  0 X^3+X^2 X^3 X^3+X^2 X^2  0 X^3+X^2  0 X^3 X^3+X^2 X^2  0 X^3+X^2  0 X^2 X^3 X^2 X^2 X^3  0 X^3+X^2 X^3+X^2 X^3  0  0 X^3 X^3  0  0  0 X^3 X^2 X^3+X^2 X^3+X^2 X^3  0 X^2 X^3+X^2  0 X^3+X^2 X^2 X^2 X^3+X^2 X^3  0 X^3 X^2
 0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0 X^3  0 X^3 X^3 X^3  0 X^3  0 X^3  0  0  0 X^3  0 X^3 X^3 X^3  0  0 X^3 X^3  0  0  0 X^3  0 X^3 X^3  0  0  0 X^3  0 X^3 X^3 X^3  0  0  0  0 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+111x^50+4x^51+157x^52+128x^53+400x^54+504x^55+396x^56+128x^57+93x^58+4x^59+69x^60+30x^62+16x^64+6x^66+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 20.7 seconds.