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

generates a code of length 51 over Z2[X]/(X^4) who´s minimum homogenous weight is 45.

Homogenous weight enumerator: w(x)=1x^0+444x^45+1854x^46+3304x^47+5419x^48+7408x^49+9346x^50+10226x^51+9360x^52+7382x^53+5483x^54+2994x^55+1400x^56+536x^57+248x^58+82x^59+36x^60+6x^61+5x^62+2x^63

The gray image is a linear code over GF(2) with n=408, k=16 and d=180.
This code was found by Heurico 1.16 in 35.6 seconds.