The generator matrix
1 1 1 1 1
0 X 0 0 0
0 0 X 0 0
0 0 0 X 0
0 0 0 0 X
generates a code of length 5 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 4.
Homogenous weight enumerator: w(x)=1x^0+15x^4+90x^8+270x^12+3072x^15+405x^16+243x^20
The gray image is a linear code over GF(4) with n=20, k=6 and d=4.
This code was found by Heurico 1.16 in 0.00164 seconds.