The generator matrix

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

generates a code of length 97 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 288.

Homogenous weight enumerator: w(x)=1x^0+528x^288+192x^292+288x^296+12x^304+3x^320

The gray image is a linear code over GF(4) with n=388, k=5 and d=288.
This code was found by Heurico 1.16 in 0.172 seconds.