The generator matrix

 1  1  1  1  X  X  X X^2  X  X
 0 X^3+X^2  0 X^2  0 X^2 X^3+X^2 X^3 X^3 X^3
 0  0 X^3+X^2 X^2 X^3+X^2 X^2  0 X^2 X^2 X^3
 0  0  0 X^3  0  0 X^3  0 X^3  0
 0  0  0  0 X^3  0 X^3  0 X^3 X^3
 0  0  0  0  0 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 6.

Homogenous weight enumerator: w(x)=1x^0+55x^6+48x^7+106x^8+464x^9+699x^10+464x^11+112x^12+48x^13+45x^14+5x^16+1x^18

The gray image is a linear code over GF(2) with n=80, k=11 and d=24.
This code was found by Heurico 1.16 in 0.031 seconds.