The generator matrix

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

generates a code of length 53 over Z2[X]/(X^4) who´s minimum homogenous weight is 47.

Homogenous weight enumerator: w(x)=1x^0+498x^47+1994x^48+3404x^49+5526x^50+7352x^51+8967x^52+10302x^53+9071x^54+7352x^55+5421x^56+2988x^57+1662x^58+660x^59+193x^60+74x^61+51x^62+10x^63+8x^64+2x^66

The gray image is a linear code over GF(2) with n=424, k=16 and d=188.
This code was found by Heurico 1.16 in 28.2 seconds.