The generator matrix

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

generates a code of length 15 over Z2[X]/(X^4) who´s minimum homogenous weight is 14.

Homogenous weight enumerator: w(x)=1x^0+15x^14+224x^15+15x^16+1x^30

The gray image is a linear code over GF(2) with n=120, k=8 and d=56.
As d=58 is an upper bound for linear (120,8,2)-codes, this code is optimal over Z2[X]/(X^4) for dimension 8.
This code was found by Heurico 1.16 in 3.81e-009 seconds.