The generator matrix

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

generates a code of length 55 over Z2[X]/(X^4) who´s minimum homogenous weight is 54.

Homogenous weight enumerator: w(x)=1x^0+100x^54+64x^55+74x^56+12x^58+5x^64

The gray image is a linear code over GF(2) with n=440, k=8 and d=216.
This code was found by Heurico 1.16 in 0.125 seconds.