The generator matrix

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

generates a code of length 23 over Z2[X]/(X^2) who´s minimum homogenous weight is 16.

Homogenous weight enumerator: w(x)=1x^0+415x^16+490x^18+1419x^20+1628x^22+2022x^24+1296x^26+658x^28+164x^30+90x^32+6x^34+3x^36

The gray image is a linear code over GF(2) with n=46, k=13 and d=16.
As d=16 is an upper bound for linear (46,13,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 13.
This code was found by Heurico 1.16 in 59.1 seconds.