The generator matrix

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

generates a code of length 55 over Z2[X]/(X^4) who´s minimum homogenous weight is 51.

Homogenous weight enumerator: w(x)=1x^0+462x^51+866x^52+1358x^53+1056x^54+1382x^55+982x^56+776x^57+392x^58+418x^59+240x^60+182x^61+40x^62+26x^63+5x^64+4x^65+2x^68

The gray image is a linear code over GF(2) with n=440, k=13 and d=204.
This code was found by Heurico 1.16 in 4.52 seconds.