The generator matrix

 1  1  1  1  0 X^3+X^2  1  1  X  1
 0  X  0  X  X  X X^2+X X^3+X^2 X^3+X^2+X  0
 0  0  X  X X^3+X^2 X^2+X X^2+X  0 X^2  0
 0  0  0 X^3  0  0 X^3  0 X^3  0
 0  0  0  0 X^3 X^3 X^3 X^3 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+31x^6+66x^7+305x^8+896x^9+1506x^10+892x^11+298x^12+64x^13+31x^14+2x^15+4x^16

The gray image is a linear code over GF(2) with n=80, k=12 and d=24.
This code was found by Heurico 1.16 in 0.016 seconds.