The generator matrix

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

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+61x^52+120x^54+640x^55+155x^56+40x^58+6x^60+1x^108

The gray image is a linear code over GF(2) with n=440, k=10 and d=208.
This code was found by Heurico 1.16 in 3.42 seconds.