The generator matrix

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

generates a code of length 4 over Z2[X]/(X^4) who´s minimum homogenous weight is 2.

Homogenous weight enumerator: w(x)=1x^0+68x^2+352x^3+1206x^4+352x^5+68x^6+1x^8

The gray image is a linear code over GF(2) with n=32, k=11 and d=8.
This code was found by Heurico 1.16 in 0 seconds.