The generator matrix

 1  0  0  1  1  1  0 X^3+X^2 X^3+X^2 X^3+X^2  1  1  1  1 X^2+X  X  1 X^3+X  1 X^2+X  1  1 X^3+X  1  1  X  1 X^3+X^2  1  1  1 X^2  X  1 X^2+X  1  1  1 X^3+X^2+X  1  1  1  1 X^3+X^2  1  1  1 X^2  1  1  1  1 X^2+X  0  1
 0  1  0  0 X^2+1 X^3+X^2+1  1  X  1  1 X^2+1 X^2+1 X^3+X^2 X^2 X^2 X^2+X X^3+X^2+X+1  1 X^3+X  1 X^3+X+1 X^2+X  1 X+1 X^3+X^2+X  1 X^3+X^2+X  1 X^3+X+1 X^3 X^2+X  1 X^3 X+1  1 X^3 X^3+X^2+1 X^3+1  1 X^3+X^2+X+1 X^3 X^3+X^2+X+1 X^3+X^2  1 X^3+X X^3+X^2+1 X^3  1 X^3+X^2+X+1 X^3+1 X^2+X+1 X^3+X^2  1 X^3+X^2+X X^3+X^2+X
 0  0  1 X+1 X^2+X+1 X^2 X^2+X+1  1  X X^3+1 X^3+1 X^3+X^2+X  X X^3+X^2+1  1  1 X^3+1 X^3+1  0 X^3 X^3+X^2 X+1 X^3+X^2+X+1 X^3+X^2+X+1 X^3+X X^3+X^2+X X^3+X^2+1 X^3+X X^3+X X^2+1 X^2+X+1 X^2  1  X X^3+1 X^2+X X^3 X^2+X+1 X^3+X^2+X  1 X^3+X^2+X+1 X^2 X^2  1  0 X^3+X^2+X X^3 X^3 X^3+X  X X^3+X+1 X^2+X X^2+1  1 X+1
 0  0  0 X^2 X^2  0 X^2 X^3+X^2 X^2 X^3 X^3 X^3+X^2 X^2  0 X^2  0  0 X^2 X^3 X^3 X^3  0 X^3 X^2 X^2 X^2 X^3  0 X^3+X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^3  0  0  0 X^3+X^2 X^3+X^2 X^3 X^2 X^3+X^2 X^3+X^2 X^3 X^2 X^2  0 X^2 X^2 X^2 X^3 X^3 X^3+X^2 X^2 X^2 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+282x^50+1036x^51+1647x^52+2208x^53+2152x^54+2316x^55+1981x^56+2024x^57+1242x^58+748x^59+388x^60+216x^61+88x^62+28x^63+14x^64+4x^66+7x^68+2x^72

The gray image is a linear code over GF(2) with n=440, k=14 and d=200.
This code was found by Heurico 1.16 in 3.03 seconds.