The generator matrix

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

generates a code of length 13 over Z2[X]/(X^2) who´s minimum homogenous weight is 5.

Homogenous weight enumerator: w(x)=1x^0+76x^5+259x^6+630x^7+1515x^8+3028x^9+5136x^10+7612x^11+9508x^12+10108x^13+9374x^14+7528x^15+5235x^16+3100x^17+1480x^18+612x^19+252x^20+72x^21+7x^22+2x^23+1x^24

The gray image is a linear code over GF(2) with n=26, k=16 and d=5.
As d=5 is an upper bound for linear (26,16,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 16.
This code was found by Heurico 1.11 in 0.972 seconds.