The generator matrix

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

generates a code of length 44 over Z2[X]/(X^2) who´s minimum homogenous weight is 44.

Homogenous weight enumerator: w(x)=1x^0+6x^44+16x^45+6x^46+1x^48+2x^50

The gray image is a linear code over GF(2) with n=88, k=5 and d=44.
As d=44 is an upper bound for linear (88,5,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 0.0095 seconds.