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

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

Homogenous weight enumerator: w(x)=1x^0+20x^53+4x^54+4x^55+3x^56

The gray image is a linear code over GF(2) with n=416, k=5 and d=212.
This code was found by Heurico 1.16 in 0.062 seconds.