The generator matrix

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

generates a code of length 44 over Z2[X]/(X^4) who´s minimum homogenous weight is 39.

Homogenous weight enumerator: w(x)=1x^0+158x^39+837x^40+1412x^41+1999x^42+2654x^43+2639x^44+2528x^45+1916x^46+1102x^47+669x^48+264x^49+99x^50+70x^51+25x^52+4x^53+2x^54+3x^56+2x^60

The gray image is a linear code over GF(2) with n=352, k=14 and d=156.
This code was found by Heurico 1.16 in 2.13 seconds.