The generator matrix

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

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+7x^40+72x^41+50x^42+172x^43+475x^44+148x^45+24x^46+20x^47+12x^48+36x^49+5x^50+1x^52+1x^82

The gray image is a linear code over GF(2) with n=352, k=10 and d=160.
This code was found by Heurico 1.16 in 0.062 seconds.