The generator matrix

 1  0  0  0  1
 0  1  0  0 16
 0  0  1  0 14
 0  0  0  1 14

generates a code of length 5 over Z27 who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+240x^4+5060x^6+1440x^7+61560x^8+20180x^9+297720x^10+123120x^11+20190x^12+1920x^13+10x^15

The gray image is a code over GF(3) with n=45, k=12 and d=12.
This code was found by Heurico 1.16 in 5.18 seconds.