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  1  1  1  X  1  1  1  1  0 X^2  0  X  0  1
 0  X X^3+X^2 X^2+X  0 X^2+X X^3+X^2 X^3+X  0 X^3+X X^3+X^2 X^2+X X^2 X^3+X^2+X  0 X^3+X X^3  X X^3+X^2 X^2+X  0 X^3+X X^2+X X^3 X^3+X X^3+X^2 X^2 X^2+X  0 X^3+X^2+X X^3+X^2 X^3+X  0 X^3 X^2+X X^3+X^2+X X^3+X^2 X^3+X X^2  X X^3+X  X X^3+X^2 X^2 X^2+X X^3+X^2 X^3+X^2 X^2 X^2  X X^3+X^2 X^2 X^2+X  X  0
 0  0 X^3  0  0  0  0  0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0 X^3 X^3  0 X^3  0  0 X^3 X^3  0 X^3 X^3 X^3  0  0  0  0 X^3  0  0 X^3 X^3  0  0 X^3 X^3  0 X^3 X^3  0 X^3  0 X^3  0  0  0
 0  0  0 X^3  0  0  0  0  0  0 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0 X^3  0 X^3 X^3  0 X^3 X^3  0 X^3  0  0 X^3  0 X^3 X^3 X^3  0 X^3 X^3  0  0  0  0  0 X^3  0 X^3  0  0 X^3  0  0 X^3
 0  0  0  0 X^3  0  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0 X^3  0 X^3 X^3  0  0 X^3 X^3  0  0 X^3 X^3  0  0  0 X^3  0  0 X^3  0 X^3 X^3 X^3 X^3 X^3  0  0  0  0 X^3  0 X^3  0  0 X^3  0  0  0 X^3
 0  0  0  0  0 X^3 X^3 X^3 X^3  0  0  0 X^3 X^3 X^3 X^3  0  0 X^3  0 X^3  0 X^3 X^3 X^3 X^3  0  0  0 X^3  0 X^3 X^3 X^3 X^3 X^3  0  0 X^3  0 X^3 X^3  0 X^3 X^3  0 X^3 X^3  0 X^3 X^3 X^3  0 X^3  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+27x^50+186x^51+118x^52+96x^53+502x^54+210x^55+494x^56+96x^57+100x^58+174x^59+26x^60+10x^62+6x^63+1x^64+1x^98

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.