The generator matrix

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

generates a code of length 51 over Z2[X]/(X^4) who´s minimum homogenous weight is 44.

Homogenous weight enumerator: w(x)=1x^0+205x^44+1340x^45+3184x^46+6464x^47+10260x^48+15168x^49+18315x^50+20920x^51+18767x^52+15380x^53+10483x^54+6130x^55+2561x^56+1212x^57+441x^58+180x^59+26x^60+18x^61+9x^62+2x^63+2x^64+2x^68+2x^69

The gray image is a linear code over GF(2) with n=408, k=17 and d=176.
This code was found by Heurico 1.16 in 98.8 seconds.