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  X  1  1  1  1  1  1  1  1  1  1  1  1  1  X  1  1  1  1  X  X  1  X  1  1  1  1  1  1  1  X  1  1
 0  X  0  0  0  0  0  0  0  0  0 a*X a^2*X a*X  X a*X a^2*X a*X  X  0 a*X  0 a*X  X  X a^2*X a*X  X a*X a*X a*X a^2*X  0 a*X a*X a^2*X a^2*X a*X  0  X  X  0 a*X  X a*X  X  0 a*X  0 a^2*X a^2*X  X  X a*X  0 a^2*X  X a^2*X a^2*X a^2*X  X a*X a*X a*X  X  X a*X a*X  X a^2*X  X  X a*X  0 a*X a^2*X a*X  X  0  X a^2*X  0  0 a*X  X  0 a^2*X  X  0  X  X a^2*X  X  0  0  X  0  0
 0  0  X  0  0  0  0  X  X  X a*X  0 a*X a*X a*X a^2*X  0 a^2*X a^2*X  X  X a*X  0  X  0 a^2*X  X  0  X a*X a^2*X a*X  0 a^2*X  X  0 a*X  0  X  X a*X a*X a*X a*X a^2*X  X a*X  X a*X  0 a*X  X a^2*X a*X  X  0 a*X a^2*X  X  0 a*X a^2*X  X a*X  X  X  X  X a*X a^2*X  X  X a*X a^2*X a^2*X a*X a*X  X  0 a^2*X  X  0 a^2*X  X a^2*X  X  0  0  0  0 a^2*X a^2*X a*X a*X  0 a*X a*X  0
 0  0  0  X  0  0  X a^2*X a*X a*X a*X  0 a^2*X a*X a*X  X a^2*X a*X a*X  0 a^2*X  X a^2*X  X a*X  0  X  X  X  X  X  X  0  0  X a^2*X  0  X a*X a*X a*X a*X a^2*X a^2*X a*X  0  0  0  X  0  0  X  X  X  0 a^2*X a*X  X a*X  0  X a*X a^2*X a*X a*X  0  0  X  X a^2*X  0  0 a^2*X a*X a^2*X  X  X a*X a*X  0  0 a^2*X a*X a*X  0 a*X  0 a^2*X  0 a^2*X a^2*X a^2*X a^2*X a*X  0  0 a*X  0
 0  0  0  0  X  0 a^2*X  0  X a*X a^2*X a^2*X  X  0  0 a*X  0 a*X  0  X a*X  X  0 a*X  X  0 a*X a*X a^2*X a^2*X a^2*X a*X a*X a*X a*X a^2*X  X a*X  X a*X  0  X a^2*X a*X a*X a*X a^2*X  X a*X a^2*X  0  X a^2*X a^2*X  X  X a^2*X a^2*X a^2*X a*X a*X a^2*X a*X  X a*X a^2*X a^2*X  0 a^2*X a^2*X  0 a^2*X  X a^2*X  0  X  X  X  0 a^2*X  0 a*X  0  X a*X a*X  0  0 a^2*X a*X  X  X  X  0  X a^2*X  X  0
 0  0  0  0  0  X  X  X a^2*X  X  X a*X  X  0 a*X a^2*X a^2*X a*X a*X a*X a*X a^2*X  X a*X a*X a*X a*X a^2*X a^2*X a*X a^2*X a*X  X a*X  0 a^2*X  0  0 a*X  X  X  0 a^2*X a^2*X  X  0 a^2*X a^2*X a^2*X  0 a^2*X a*X a*X  X a^2*X  X a*X a*X  X a*X  0 a*X  X  X a*X  X a*X a*X a*X  0  X  X  0  0  X  X  X  X a*X a^2*X  X a*X  X a*X  X  X  X a*X  0  0  X a*X  X a*X  X  X a^2*X a^2*X

generates a code of length 98 over F4[X]/(X^2) who´s minimum homogenous weight is 268.

Homogenous weight enumerator: w(x)=1x^0+84x^268+288x^272+375x^276+510x^280+801x^284+1872x^288+4047x^292+4656x^296+2481x^300+303x^304+240x^308+168x^312+177x^316+114x^320+87x^324+81x^328+45x^332+27x^336+15x^340+9x^344+3x^368

The gray image is a linear code over GF(4) with n=392, k=7 and d=268.
This code was found by Heurico 1.16 in 4.79 seconds.