The generator matrix

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

generates a code of length 55 over Z2[X]/(X^4) who´s minimum homogenous weight is 53.

Homogenous weight enumerator: w(x)=1x^0+76x^53+42x^54+16x^55+42x^56+64x^57+4x^58+4x^60+4x^61+2x^62+1x^64

The gray image is a linear code over GF(2) with n=440, k=8 and d=212.
This code was found by Heurico 1.16 in 0.312 seconds.