The generator matrix

 1  0  1  1  1  1  1  1  1  X  1  1  1  1 aX  1  1  1  1 (a+1)X  1  1  1  1  0  1  1  1  1  X  1  1  1  1 aX  1  1  1  1 (a+1)X  1  1  1  1  0  1  1  1  1  X  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  0  1  1  1  1  1 aX (a+1)X  X aX (a+1)X  0  1  1  1  1  1  1  X  X  1  1  1  X  1  1
 0  1 (a+1)X+1  a (a+1)X+a+1  X aX+1 X+a aX+a+1  1 aX X+1 aX+a X+a+1  1 (a+1)X  1 (a+1)X+a a+1  1  0 (a+1)X+1  a (a+1)X+a+1  1  X aX+1 X+a aX+a+1  1 aX X+1 aX+a X+a+1  1 (a+1)X  1 (a+1)X+a a+1  1  0 (a+1)X+1  a (a+1)X+a+1  1  X aX+1 X+a aX+a+1  1 aX X+1 aX+a X+a+1 (a+1)X (a+1)X+1 aX+1  a X+a (a+1)X+a+1 aX+a+1  0  X (a+1)X+1 aX+1  a (a+1)X+a+1  0 X+a aX+a+1  X  1  1 aX+a X+a+1 aX+a X+a+1  1  1  1  1  1  X (a+1)X+a a+1 aX X+1 (a+1)X+a a+1 aX (a+1)X (a+1)X+a a+1 (a+1)X  0 aX (a+1)X

generates a code of length 97 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 291.

Homogenous weight enumerator: w(x)=1x^0+192x^291+33x^292+12x^296+12x^300+3x^304+3x^308

The gray image is a linear code over GF(4) with n=388, k=4 and d=291.
This code was found by Heurico 1.16 in 42.2 seconds.