The generator matrix

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

generates a code of length 44 over Z2[X]/(X^4) who´s minimum homogenous weight is 40.

Homogenous weight enumerator: w(x)=1x^0+284x^40+828x^41+1307x^42+1192x^43+1479x^44+1032x^45+892x^46+522x^47+382x^48+170x^49+51x^50+30x^51+13x^52+2x^53+6x^54+1x^56

The gray image is a linear code over GF(2) with n=352, k=13 and d=160.
This code was found by Heurico 1.16 in 0.969 seconds.