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  X  1  1  1  1  1  1  X  1 X^2  0  1  1  X  1  1 X^2 X^2
 0 X^3+X^2  0 X^2  0  0 X^2 X^2 X^3 X^3 X^2 X^3+X^2  0  0 X^2 X^2 X^3 X^2 X^3 X^2 X^3+X^2  0  0 X^2 X^3+X^2 X^2  0  0 X^3 X^3+X^2 X^3 X^3+X^2  0 X^3 X^3  0 X^3+X^2 X^2 X^3 X^2  0 X^3 X^3  0 X^2  0 X^3+X^2 X^2 X^2 X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^3+X^2  0
 0  0 X^3+X^2 X^2  0 X^3+X^2 X^3+X^2  0 X^3 X^2 X^2  0 X^3 X^3+X^2 X^3+X^2  0 X^3 X^2 X^3+X^2 X^3 X^3 X^3+X^2  0 X^3+X^2  0  0  0 X^3 X^2 X^3+X^2 X^3+X^2 X^2  0 X^2 X^3 X^3+X^2 X^3 X^3+X^2 X^3+X^2 X^2 X^3 X^3+X^2 X^3 X^3 X^2 X^2  0  0  0 X^2 X^2  0 X^2  0 X^3+X^2
 0  0  0 X^3  0  0 X^3  0  0 X^3  0 X^3 X^3  0  0 X^3 X^3 X^3  0 X^3  0 X^3  0 X^3 X^3  0 X^3 X^3 X^3  0 X^3  0  0  0  0  0 X^3  0 X^3  0  0  0 X^3 X^3 X^3 X^3  0  0 X^3  0  0 X^3 X^3 X^3  0
 0  0  0  0 X^3  0  0  0  0  0  0  0 X^3 X^3 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0 X^3  0 X^3 X^3 X^3  0  0  0  0 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3  0  0  0 X^3 X^3 X^3  0 X^3 X^3 X^3  0 X^3 X^3
 0  0  0  0  0 X^3 X^3 X^3 X^3 X^3  0 X^3  0 X^3 X^3  0 X^3 X^3  0 X^3  0  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 X^3  0 X^3  0 X^3 X^3 X^3  0 X^3 X^3  0 X^3  0 X^3  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+129x^50+16x^51+97x^52+224x^53+329x^54+544x^55+253x^56+224x^57+123x^58+16x^59+27x^60+43x^62+5x^64+16x^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 41.9 seconds.