The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  X  1  1  X  X  X  X  1  1  1  1  1  1 X^2  0  X  X X^2  0  X  X X^2 X^2 X^2 X^2  X  X X^3 X^3  1  1 X^2  1  1 X^2  1
 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^2  0 X^3 X^3 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^2 X^3+X^2 X^3+X^2  0 X^3 X^2 X^2 X^2 X^2 X^2 X^2  0 X^3 X^3  0 X^3  0  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 X^3  0  0 X^3 X^3 X^3 X^3  0 X^3  0 X^3  0  0  0  0 X^3 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+6x^52+104x^53+6x^54+1x^56+8x^57+1x^58+1x^66

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 0.094 seconds.