The generator matrix

 1  0  1  1  1  X  1  1  0  1  1  X  1  1  X  X  0  0  1  1  0  X
 0  1  1  X X+1  1  0 X+1  1  X  1  1  0  X  0  X  X  0 X+1  1  1  1
 0  0  X  0  X  0  X  0  X  X  0  X  0  0  X  X  X  X  0  0  X  X

generates a code of length 22 over Z2[X]/(X^2) who´s minimum homogenous weight is 22.

Homogenous weight enumerator: w(x)=1x^0+24x^22+6x^24+1x^32

The gray image is a linear code over GF(2) with n=44, k=5 and d=22.
As d=22 is an upper bound for linear (44,5,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 0.00581 seconds.