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.