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

generates a code of length 53 over Z2[X]/(X^4) who´s minimum homogenous weight is 52.

Homogenous weight enumerator: w(x)=1x^0+38x^52+40x^53+36x^54+8x^55+1x^56+2x^58+1x^62+1x^70

The gray image is a linear code over GF(2) with n=424, k=7 and d=208.
This code was found by Heurico 1.16 in 0.062 seconds.