The generator matrix

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

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+76x^49+219x^50+336x^51+304x^52+250x^53+274x^54+266x^55+212x^56+72x^57+11x^58+20x^59+2x^61+2x^63+2x^64+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 0.157 seconds.