The generator matrix

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

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+25x^46+156x^47+217x^48+276x^49+512x^50+634x^51+633x^52+584x^53+410x^54+260x^55+177x^56+86x^57+40x^58+26x^59+22x^60+12x^61+5x^62+12x^63+1x^64+2x^65+4x^68+1x^76

The gray image is a linear code over GF(2) with n=416, k=12 and d=184.
This code was found by Heurico 1.16 in 0.39 seconds.