The generator matrix

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

generates a code of length 53 over Z2[X]/(X^4) who´s minimum homogenous weight is 49.

Homogenous weight enumerator: w(x)=1x^0+164x^49+60x^50+232x^51+336x^52+580x^53+309x^54+160x^55+45x^56+60x^57+14x^58+56x^59+1x^60+28x^61+1x^62+1x^92

The gray image is a linear code over GF(2) with n=424, k=11 and d=196.
This code was found by Heurico 1.16 in 106 seconds.