The generator matrix

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

generates a code of length 55 over Z2[X]/(X^4) who´s minimum homogenous weight is 49.

Homogenous weight enumerator: w(x)=1x^0+594x^49+1854x^50+3808x^51+5530x^52+7522x^53+8788x^54+9690x^55+8842x^56+7562x^57+5234x^58+3490x^59+1575x^60+622x^61+244x^62+94x^63+43x^64+20x^65+16x^66+6x^67+1x^68

The gray image is a linear code over GF(2) with n=440, k=16 and d=196.
This code was found by Heurico 1.16 in 29.8 seconds.