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  X  X  X  X  X  X  X  X  X  X  X  X  X X^2  X  X  X X^3  1  1  1  1  X
 0 X^3+X^2  0 X^3+X^2  0 X^3+X^2  0 X^3+X^2  0 X^3+X^2  0 X^3+X^2  0 X^3+X^2  0 X^3+X^2 X^3 X^2 X^3 X^2 X^3 X^2 X^3 X^2 X^3 X^2 X^3 X^2 X^3 X^2 X^3 X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^2  0  0  0 X^3  0  0 X^3 X^3 X^3 X^3  0  0  0 X^3+X^2 X^3+X^2
 0  0 X^3  0  0  0 X^3  0  0 X^3  0 X^3 X^3 X^3 X^3 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 X^3 X^3 X^3 X^3  0  0 X^3 X^3  0 X^3 X^3 X^3  0  0  0 X^3  0  0  0
 0  0  0 X^3  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3  0 X^3  0  0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0  0  0  0 X^3 X^3 X^3 X^3  0  0  0 X^3 X^3  0 X^3 X^3  0 X^3  0  0  0 X^3 X^3  0 X^3
 0  0  0  0 X^3 X^3 X^3 X^3 X^3  0  0 X^3  0 X^3 X^3  0  0  0 X^3 X^3 X^3 X^3  0  0  0  0 X^3 X^3 X^3 X^3  0  0  0 X^3 X^3  0  0 X^3 X^3  0 X^3 X^3  0  0  0  0 X^3 X^3  0  0  0 X^3 X^3  0  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+37x^52+32x^53+192x^54+32x^55+180x^56+4x^58+19x^60+10x^62+2x^64+2x^70+1x^72

The gray image is a linear code over GF(2) with n=440, k=9 and d=208.
This code was found by Heurico 1.16 in 0.125 seconds.