The generator matrix

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

generates a code of length 29 over Z2[X]/(X^3) who´s minimum homogenous weight is 29.

Homogenous weight enumerator: w(x)=1x^0+20x^29+2x^30+4x^31+3x^32+2x^34

The gray image is a linear code over GF(2) with n=116, k=5 and d=58.
As d=59 is an upper bound for linear (116,5,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 5.
This code was found by Heurico 1.16 in 0.00555 seconds.