The generator matrix

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

generates a code of length 13 over Z2[X]/(X^3) who´s minimum homogenous weight is 13.

Homogenous weight enumerator: w(x)=1x^0+8x^13+5x^14+1x^16+1x^18

The gray image is a linear code over GF(2) with n=52, k=4 and d=26.
As d=27 is an upper bound for linear (52,4,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 4.
This code was found by Heurico 1.16 in 0.000438 seconds.