The generator matrix

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

generates a code of length 55 over Z2[X]/(X^4) who´s minimum homogenous weight is 56.

Homogenous weight enumerator: w(x)=1x^0+16x^56+10x^57+2x^58+2x^59+1x^60

The gray image is a linear code over GF(2) with n=440, k=5 and d=224.
This code was found by Heurico 1.16 in 0.063 seconds.