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

generates a code of length 85 over F4[X]/(X^2) who´s minimum homogenous weight is 248.

Homogenous weight enumerator: w(x)=1x^0+69x^248+66x^252+768x^255+42x^256+39x^260+15x^264+12x^268+9x^272+3x^340

The gray image is a linear code over GF(4) with n=340, k=5 and d=248.
This code was found by Heurico 1.16 in 0.109 seconds.