The generator matrix
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X X X X X X X 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 X X X X X X X X 0 0 0 0 0 0 0 1 1 X 1 1 1 1 1
0 X 0 0 0 X X X 0 0 0 X 0 X X X 0 0 0 X 0 X X X 0 0 X X 0 X X X X X X 0 0 0 0 0 0 X 0 X X X 0 0 X X 0 X X 0 X X X X 0 0 0 0 0 0 0 X 0 X X
0 0 X 0 X X X 0 0 0 X X X X 0 0 0 0 X X X X 0 0 0 X X 0 X X 0 0 0 X X X X 0 0 0 X X X X 0 0 0 X X 0 X X 0 0 0 0 X X X X 0 0 0 0 X X X X 0
0 0 0 X X 0 X X 0 X X 0 0 X X 0 0 X X 0 0 X X 0 X X 0 0 0 X X 0 X X 0 0 X X 0 X X 0 0 X X 0 X X 0 0 0 X X 0 0 X X 0 0 X X 0 X 0 X 0 0 X X
generates a code of length 69 over Z2[X]/(X^2) who´s minimum homogenous weight is 70.
Homogenous weight enumerator: w(x)=1x^0+7x^70+16x^71+7x^72+1x^78
The gray image is a linear code over GF(2) with n=138, k=5 and d=70.
As d=70 is an upper bound for linear (138,5,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 0.0672 seconds.