The generator matrix
1 0 0 0 1
0 1 0 0 a^2*X+1
0 0 1 0 a^2
0 0 0 1 1
0 0 0 0 X
generates a code of length 5 over F4[X]/(X^2) who´s minimum homogenous weight is 4.
Homogenous weight enumerator: w(x)=1x^0+15x^4+480x^6+90x^8+3840x^9+4320x^10+27150x^12+23040x^13+12960x^14+61440x^15+81045x^16+34560x^17+12960x^18+243x^20
The gray image is a linear code over GF(4) with n=20, k=9 and d=4.
This code was found by Heurico 1.16 in 2.83 seconds.