The generator matrix

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

generates a code of length 96 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 269.

Homogenous weight enumerator: w(x)=1x^0+252x^269+348x^270+168x^271+330x^272+1440x^273+1512x^274+696x^275+684x^276+2616x^277+2316x^278+1164x^279+993x^280+3696x^281+3048x^282+1404x^283+1167x^284+4392x^285+3420x^286+1464x^287+1053x^288+5064x^289+3624x^290+1716x^291+1095x^292+4092x^293+3144x^294+1332x^295+906x^296+3552x^297+2472x^298+804x^299+513x^300+1716x^301+1152x^302+360x^303+291x^304+744x^305+384x^306+84x^307+42x^308+84x^309+84x^310+24x^311+18x^312+33x^316+6x^320+9x^324+12x^328+3x^332+3x^336+6x^340+3x^344

The gray image is a linear code over GF(4) with n=384, k=8 and d=269.
This code was found by Heurico 1.16 in 35.3 seconds.