The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+40x^53+6x^54+8x^55+7x^56+2x^58

The gray image is a linear code over GF(2) with n=424, k=6 and d=212.
This code was found by Heurico 1.16 in 0.078 seconds.