The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  X  1  X  1  X  1  1  X  1  X  X  1  1  1  1  1  1  1  1  X  1  X  1  1  1  1  1  X  X  X  X  X  X  1 X^2  X
 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 X^3+X^2  0 X^3+X^2 X^3+X^2 X^3+X^2  0  0 X^3  0 X^3 X^3 X^2 X^3+X^2 X^2 X^3+X^2 X^2 X^3+X^2 X^2  0 X^2  0 X^2 X^3  0 X^3 X^3 X^3  0 X^2  0 X^2  0 X^3 X^3 X^3+X^2 X^3+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 X^3  0 X^3 X^3 X^3 X^3  0 X^3  0  0  0 X^3 X^3  0  0 X^3  0 X^3 X^3 X^3  0  0 X^3 X^3  0  0  0 X^3  0 X^3  0  0
 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 X^3  0 X^3  0  0 X^3 X^3  0  0 X^3 X^3 X^3  0 X^3 X^3  0 X^3  0  0  0 X^3 X^3  0  0 X^3  0 X^3 X^3 X^3 X^3  0  0  0 X^3  0  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+109x^52+64x^53+64x^54+14x^56+1x^60+2x^68+1x^72

The gray image is a linear code over GF(2) with n=424, k=8 and d=208.
This code was found by Heurico 1.16 in 44.2 seconds.