The generator matrix

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

generates a code of length 55 over Z2[X]/(X^4) who´s minimum homogenous weight is 47.

Homogenous weight enumerator: w(x)=1x^0+166x^47+1289x^48+3364x^49+6887x^50+11960x^51+20321x^52+29714x^53+36516x^54+40818x^55+37275x^56+30482x^57+20453x^58+11994x^59+6390x^60+2526x^61+1250x^62+480x^63+157x^64+68x^65+14x^66+6x^67+7x^68+4x^69+2x^73

The gray image is a linear code over GF(2) with n=440, k=18 and d=188.
This code was found by Heurico 1.16 in 431 seconds.