The generator matrix
1 0 1 1 1 1 1
0 1 1 a 0 0 0
0 0 (a+1)X 0 0 X 0
0 0 0 X 0 aX 0
0 0 0 0 X (a+1)X 0
generates a code of length 7 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 8.
Homogenous weight enumerator: w(x)=1x^0+30x^8+48x^9+75x^12+288x^13+165x^16+576x^17+2304x^18+270x^20+3936x^21+6912x^22+300x^24+1296x^25+183x^28
The gray image is a linear code over GF(4) with n=28, k=7 and d=8.
This code was found by Heurico 1.16 in 0.0223 seconds.