The generator matrix
1 0 0 0 1 1 1
0 1 0 0 1 a^2*X 0
0 0 1 0 a^2 1 0
0 0 0 1 1 a X
0 0 0 0 a^2*X a*X a^2*X
generates a code of length 7 over F4[X]/(X^2) who´s minimum homogenous weight is 8.
Homogenous weight enumerator: w(x)=1x^0+9x^8+48x^9+120x^10+972x^12+576x^13+480x^14+3840x^15+8403x^16+2592x^17+20112x^18+23040x^19+24912x^20+54336x^21+57696x^22+34560x^23+24963x^24+3888x^25+1464x^26+132x^28
The gray image is a linear code over GF(4) with n=28, k=9 and d=8.
This code was found by Heurico 1.16 in 4.77 seconds.