The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+40x^55+2x^56+16x^57+3x^58+1x^60+1x^62

The gray image is a linear code over GF(2) with n=440, k=6 and d=220.
This code was found by Heurico 1.16 in 0.094 seconds.