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  1  1  1  1  1  1  X  1 X^2  1  1
 0 X^3+X^2  0 X^3+X^2  0 X^3+X^2 X^3 X^3+X^2 X^2  0  0 X^3+X^2  0 X^3+X^2 X^3 X^2 X^3+X^2  0 X^3 X^2  0 X^3+X^2 X^3 X^2  0 X^3+X^2  0 X^3+X^2 X^3 X^2 X^3 X^2  0 X^3+X^2  0 X^3 X^3 X^3+X^2 X^2 X^2  0 X^3 X^3 X^3 X^3+X^2 X^2 X^3+X^2 X^2 X^3 X^2 X^3+X^2  0  0  0  0
 0  0 X^3  0  0  0 X^3  0  0  0  0  0  0 X^3  0 X^3 X^3  0  0 X^3 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3  0  0  0  0  0  0 X^3 X^3  0 X^3 X^3 X^3  0  0  0 X^3 X^3  0  0 X^3 X^3  0 X^3  0
 0  0  0 X^3  0  0  0  0 X^3 X^3 X^3  0  0  0 X^3  0  0  0 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3  0 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0 X^3 X^3  0  0 X^3  0 X^3  0  0 X^3 X^3 X^3 X^3 X^3  0
 0  0  0  0 X^3  0 X^3 X^3 X^3 X^3  0 X^3  0 X^3  0  0  0 X^3 X^3 X^3  0  0 X^3 X^3  0 X^3 X^3  0  0 X^3 X^3  0 X^3 X^3  0  0 X^3 X^3  0  0 X^3  0  0 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0
 0  0  0  0  0 X^3  0 X^3  0 X^3 X^3  0 X^3 X^3  0  0 X^3 X^3  0  0  0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0 X^3 X^3 X^3 X^3  0 X^3  0  0  0 X^3 X^3  0 X^3 X^3 X^3  0  0  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+15x^50+10x^51+27x^52+66x^53+85x^54+620x^55+91x^56+60x^57+20x^58+10x^59+9x^60+2x^61+7x^62+1x^106

The gray image is a linear code over GF(2) with n=440, k=10 and d=200.
This code was found by Heurico 1.16 in 0.156 seconds.