The generator matrix

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

generates a code of length 55 over Z2[X]/(X^4) who´s minimum homogenous weight is 52.

Homogenous weight enumerator: w(x)=1x^0+700x^52+444x^53+1109x^54+280x^55+612x^56+176x^57+376x^58+104x^59+212x^60+20x^61+50x^62+11x^64+1x^70

The gray image is a linear code over GF(2) with n=440, k=12 and d=208.
This code was found by Heurico 1.16 in 6.58 seconds.