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

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+18x^52+104x^53+45x^54+688x^55+45x^56+104x^57+18x^58+1x^110

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 0.094 seconds.