The generator matrix

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

generates a code of length 10 over Z2[X]/(X^4) who´s minimum homogenous weight is 8.

Homogenous weight enumerator: w(x)=1x^0+67x^8+288x^9+312x^10+288x^11+66x^12+2x^16

The gray image is a linear code over GF(2) with n=80, k=10 and d=32.
This code was found by Heurico 1.16 in 3.62e-008 seconds.