The generator matrix

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

generates a code of length 44 over Z2[X]/(X^4) who´s minimum homogenous weight is 36.

Homogenous weight enumerator: w(x)=1x^0+16x^36+550x^37+2028x^38+4498x^39+11092x^40+18054x^41+31349x^42+39954x^43+46731x^44+39550x^45+32486x^46+18700x^47+10519x^48+3974x^49+1737x^50+570x^51+209x^52+92x^53+16x^54+6x^55+6x^56+4x^57+2x^60

The gray image is a linear code over GF(2) with n=352, k=18 and d=144.
This code was found by Heurico 1.16 in 314 seconds.