The generator matrix
1 1 1 1 1 1 X X 1 X X^2 1 X^2 X
0 X^3 0 X^3 0 X^3 X^3 0 0 X^3 X^3 X^3 0 0
generates a code of length 14 over Z2[X]/(X^4) who´s minimum homogenous weight is 14.
Homogenous weight enumerator: w(x)=1x^0+24x^14+7x^16
The gray image is a linear code over GF(2) with n=112, k=5 and d=56.
As d=56 is an upper bound for linear (112,5,2)-codes, this code is optimal over Z2[X]/(X^4) for dimension 5.
This code was found by Heurico 1.16 in 3.62e-008 seconds.