The generator matrix
1 1 1 1 1 1 1 1 X X X X X^2
0 X^3 0 0 0 X^3 X^3 X^3 0 0 X^3 0 X^3
0 0 X^3 0 X^3 X^3 X^3 0 0 X^3 X^3 X^3 X^3
0 0 0 X^3 X^3 0 X^3 X^3 X^3 X^3 0 0 0
generates a code of length 13 over Z2[X]/(X^4) who´s minimum homogenous weight is 12.
Homogenous weight enumerator: w(x)=1x^0+36x^12+64x^13+16x^14+11x^16
The gray image is a linear code over GF(2) with n=104, k=7 and d=48.
As d=51 is an upper bound for linear (104,7,2)-codes, this code is optimal over Z2[X]/(X^4) for dimension 7.
This code was found by Heurico 1.16 in -6.48e-008 seconds.