The generator matrix

 1  0  1  1  1 X^2+X  1  1 X^3+X^2  1  1 X^3+X  1  1  0  1  1 X^2+X  1  1 X^3+X^2  1  1 X^3+X  1  1  0  1  1 X^2+X X^3+X  1  1  1  1  0  X  1  1 X^3  1 X^3+X^2  1  1
 0  1 X+1 X^2+X X^2+1  1 X^3+X^2+X+1 X^3+X^2  1 X^3+X X^3+1  1  0 X+1  1 X^2+X X^2+1  1 X^3+X^2 X^3+X^2+X+1  1 X^3+X X^3+1  1  0 X+1  1 X^3+X^2+X+1 X^2+X  1  1 X^2+1  0 X^2+X X^2+1  1 X^2+X X^3+X^2+1 X^3  1 X^3+X^2  1 X^3+1  0
 0  0 X^3  0  0  0  0 X^3 X^3 X^3 X^3 X^3  0  0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0  0 X^3 X^3 X^3  0  0 X^3  0 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3
 0  0  0 X^3  0 X^3 X^3 X^3 X^3  0 X^3  0 X^3  0 X^3  0 X^3 X^3  0 X^3  0 X^3  0  0  0  0 X^3  0  0 X^3 X^3  0  0  0  0 X^3 X^3 X^3 X^3  0  0 X^3  0 X^3
 0  0  0  0 X^3  0 X^3 X^3 X^3 X^3  0 X^3  0 X^3  0  0 X^3 X^3 X^3  0 X^3 X^3  0  0 X^3  0 X^3 X^3  0  0 X^3  0  0 X^3 X^3  0  0 X^3  0  0  0  0 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+72x^40+200x^41+227x^42+392x^43+304x^44+360x^45+216x^46+184x^47+69x^48+16x^49+2x^50+2x^56+3x^58

The gray image is a linear code over GF(2) with n=352, k=11 and d=160.
This code was found by Heurico 1.16 in 0.063 seconds.