The generator matrix

 1  0  1  1  1 X^2+X  1  1 X^3+X^2  1  1 X^3+X  1  1  0  1  1 X^2+X  1  1 X^3+X  1  1 X^3+X^2  1  1  1  1  1  1  1  1 X^3+X  1  1  1  1  1  1  1  1  0 X^3+X^2  X  1  1  1  1  1  1  1 X^3+X  1
 0  1 X+1 X^2+X X^2+1  1 X^3+X^2+X+1 X^3+X^2  1 X^3+X X^3+1  1  0 X+1  1 X^2+X X^2+1  1 X^3+X^2 X^3+X^2+X+1  1 X^3+X X^3+1  1  0 X^2+X X+1 X^3+1 X^3+X^2 X^3+X X^3+X^2+X+1 X^2+1  1  0 X^2+X X^3 X^3+X^2+X X+1 X^3+X+1 X^2+1 X^3+X^2+1  1  1 X^3+X^2+X X^2+X+1 X^2+1 X^3+X^2 X^3+X X^2 X+1 X^2+X  1  0
 0  0 X^3  0  0  0  0 X^3 X^3 X^3 X^3 X^3  0  0  0 X^3 X^3 X^3  0  0 X^3 X^3 X^3  0  0  0 X^3 X^3 X^3  0 X^3  0  0 X^3  0 X^3 X^3  0 X^3  0  0  0 X^3  0 X^3 X^3  0  0 X^3  0 X^3 X^3  0
 0  0  0 X^3  0 X^3 X^3 X^3 X^3  0 X^3  0  0  0 X^3  0  0 X^3 X^3 X^3  0 X^3 X^3  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3  0  0 X^3 X^3 X^3  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3 X^3
 0  0  0  0 X^3  0 X^3 X^3 X^3 X^3  0 X^3 X^3  0 X^3  0 X^3  0 X^3  0  0  0 X^3 X^3  0  0  0  0  0 X^3 X^3  0 X^3 X^3 X^3  0 X^3 X^3 X^3 X^3  0  0 X^3  0  0 X^3  0 X^3  0  0 X^3  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+88x^49+205x^50+300x^51+274x^52+328x^53+314x^54+272x^55+122x^56+96x^57+38x^58+4x^59+2x^60+2x^62+1x^66+1x^80

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 0.109 seconds.