The generator matrix

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

generates a code of length 10 over Z2[X]/(X^4) who´s minimum homogenous weight is 6.

Homogenous weight enumerator: w(x)=1x^0+73x^6+154x^7+1067x^8+3424x^9+6914x^10+3444x^11+1106x^12+144x^13+53x^14+2x^15+2x^16

The gray image is a linear code over GF(2) with n=80, k=14 and d=24.
This code was found by Heurico 1.16 in 0.203 seconds.