The generator matrix
1 0 1 1 1 0 1 1 X 1 1 1 1 aX 1 1 1 (a+1)X 1 1 1 1 1 X 1 1 aX
0 1 (a+1)X+1 0 (a+1)X+1 1 X aX+1 1 a X+a aX X+1 1 aX+a (a+1)X 1 1 (a+1)X+a+1 aX+a+1 X+a+1 X aX+1 1 aX X+1 1
generates a code of length 27 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 80.
Homogenous weight enumerator: w(x)=1x^0+189x^80+60x^84+6x^96
The gray image is a linear code over GF(4) with n=108, k=4 and d=80.
As d=80 is an upper bound for linear (108,4,4)-codes, this code is optimal over F4[X,sigma]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 0.172 seconds.