The generator matrix

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

generates a code of length 22 over Z2[X]/(X^3) 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=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^3) for dimension 5.
This code was found by Heurico 1.16 in 0.00211 seconds.