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

generates a code of length 52 over Z2[X]/(X^4) who´s minimum homogenous weight is 48.

Homogenous weight enumerator: w(x)=1x^0+79x^48+228x^49+269x^50+396x^51+322x^52+284x^53+188x^54+88x^55+66x^56+56x^57+20x^58+36x^59+12x^60+2x^62+1x^82

The gray image is a linear code over GF(2) with n=416, k=11 and d=192.
This code was found by Heurico 1.16 in 0.171 seconds.