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

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+143x^50+228x^51+347x^52+406x^53+674x^54+684x^55+565x^56+418x^57+264x^58+128x^59+122x^60+38x^61+32x^62+16x^63+20x^64+2x^65+6x^66+1x^68+1x^90

The gray image is a linear code over GF(2) with n=440, k=12 and d=200.
This code was found by Heurico 1.16 in 0.359 seconds.