The generator matrix
1 1 1 1 1 1 1 1 X 0 X 0 X^2
0 X 0 X^2+X X^2 X^2+X X^2 X X^2+X X X^2+X X 0
0 0 X^2 X^2 X^2 0 0 X^2 0 X^2 X^2 0 0
generates a code of length 13 over Z2[X]/(X^3) who´s minimum homogenous weight is 12.
Homogenous weight enumerator: w(x)=1x^0+36x^12+16x^14+11x^16
The gray image is a linear code over GF(2) with n=52, k=6 and d=24.
As d=25 is an upper bound for linear (52,6,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 6.
This code was found by Heurico 1.16 in 0.000663 seconds.