The generator matrix

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

generates a code of length 96 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 280.

Homogenous weight enumerator: w(x)=1x^0+60x^280+54x^284+192x^285+69x^288+576x^289+21x^292+18x^296+15x^300+6x^304+3x^308+6x^312+3x^380

The gray image is a linear code over GF(4) with n=384, k=5 and d=280.
This code was found by Heurico 1.16 in 0.171 seconds.