The generator matrix

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

generates a code of length 55 over Z2[X]/(X^4) who´s minimum homogenous weight is 51.

Homogenous weight enumerator: w(x)=1x^0+172x^51+603x^52+572x^53+571x^54+458x^55+603x^56+454x^57+299x^58+150x^59+127x^60+42x^61+24x^62+4x^63+9x^64+4x^65+1x^70+1x^72+1x^74

The gray image is a linear code over GF(2) with n=440, k=12 and d=204.
This code was found by Heurico 1.16 in 0.297 seconds.