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

generates a code of length 52 over Z2[X]/(X^4) who´s minimum homogenous weight is 46.

Homogenous weight enumerator: w(x)=1x^0+524x^46+1804x^47+3185x^48+5884x^49+7332x^50+9248x^51+9626x^52+9728x^53+7399x^54+5360x^55+2850x^56+1622x^57+614x^58+240x^59+70x^60+28x^61+11x^62+4x^63+4x^64+2x^65

The gray image is a linear code over GF(2) with n=416, k=16 and d=184.
This code was found by Heurico 1.16 in 34.5 seconds.