The generator matrix

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

generates a code of length 96 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 272.

Homogenous weight enumerator: w(x)=1x^0+264x^272+1710x^276+2283x^280+2550x^284+2892x^288+2688x^292+2355x^296+1044x^300+468x^304+33x^308+21x^312+24x^316+21x^320+6x^324+9x^328+3x^332+6x^336+3x^348+3x^356

The gray image is a linear code over GF(4) with n=384, k=7 and d=272.
This code was found by Heurico 1.16 in 2.33 seconds.