The generator matrix

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

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+12x^54+88x^55+14x^56+4x^58+8x^59+1x^64

The gray image is a linear code over GF(2) with n=440, k=7 and d=216.
This code was found by Heurico 1.16 in 0.125 seconds.