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  X  1  1 X^2  X  X  X  0 X^2 X^3  X X^2  0  X
 0 X^3+X^2  0 X^2  0  0 X^2 X^2 X^3 X^3 X^2 X^3+X^2 X^3 X^3+X^2  0 X^3+X^2 X^2 X^3  0 X^2 X^3 X^3+X^2 X^3 X^3+X^2 X^3 X^3 X^3+X^2 X^3+X^2 X^3 X^3+X^2 X^3+X^2 X^3 X^3+X^2 X^3+X^2 X^2 X^2 X^3+X^2 X^2 X^2 X^2 X^3+X^2 X^3+X^2 X^2 X^2
 0  0 X^3+X^2 X^2  0 X^3+X^2 X^3+X^2  0 X^3 X^2 X^2  0 X^3+X^2 X^2  0 X^3 X^2 X^2 X^3  0 X^3 X^2 X^2  0  0 X^2  0 X^2 X^3+X^2 X^3 X^3 X^3 X^3+X^2  0 X^2 X^3+X^2  0 X^3 X^2 X^3+X^2  0  0 X^3 X^3
 0  0  0 X^3  0  0 X^3  0  0 X^3  0  0 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3 X^3  0  0  0 X^3  0 X^3 X^3 X^3  0
 0  0  0  0 X^3  0  0  0  0  0  0 X^3 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0 X^3  0 X^3 X^3 X^3  0 X^3  0  0 X^3 X^3  0 X^3  0 X^3  0 X^3  0  0  0
 0  0  0  0  0 X^3 X^3 X^3 X^3 X^3  0 X^3  0 X^3 X^3  0 X^3 X^3  0 X^3 X^3  0 X^3 X^3 X^3  0  0  0  0  0  0  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3  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+330x^40+128x^41+384x^43+384x^44+384x^45+128x^47+294x^48+14x^56+1x^64

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 37.4 seconds.