The generator matrix

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

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+126x^50+240x^51+359x^52+496x^53+569x^54+576x^55+551x^56+512x^57+295x^58+208x^59+104x^60+16x^61+30x^62+7x^64+2x^66+1x^68+1x^70+1x^72+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 0.328 seconds.