The generator matrix

 1  1  1  1  1  1  1  1  X  X  X  X  X  X  X  X
 0 X^3+X^2  0 X^3+X^2 X^3 X^2 X^3 X^2 X^3+X^2 X^3+X^2  0 X^3 X^2 X^2  0 X^3
 0  0 X^3 X^3 X^3 X^3  0  0  0 X^3 X^3 X^3 X^3  0  0  0

generates a code of length 16 over Z2[X]/(X^4) who´s minimum homogenous weight is 16.

Homogenous weight enumerator: w(x)=1x^0+125x^16+2x^24

The gray image is a linear code over GF(2) with n=128, k=7 and d=64.
As d=64 is an upper bound for linear (128,7,2)-codes, this code is optimal over Z2[X]/(X^4) for dimension 7.
This code was found by Heurico 1.16 in -3.24e-008 seconds.