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  X  X  X  X  X  X  X  X  X  X  X  X  X  X  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  2  0  2  0  2  0  2  0  X  X  X  X  2  2  2  2  2  2  2  2  2  2  X  X  X  X 2X 2X 2X 2X  1  1  2  X  1  1
 0 2X+2  0 2X+2  0 2X+2  0 2X+2 2X  2 2X  2 2X  2 2X  2  0 2X+2  0 2X+2  0 2X+2  0 2X+2 2X  2 2X  2 2X  2 2X  2 2X+2 2X+2 2X+2 2X+2  2  2  0 2X  2  2  0 2X  0 2X  0 2X+2  0 2X+2  0 2X+2  0 2X+2 2X  2 2X  2 2X  2 2X  2 2X+2  2 2X+2  2 2X+2  2 2X+2  2 2X+2 2X+2 2X+2 2X+2  0  0  0 2X 2X 2X  2  2  2  2  2  2  2  2  2  2  2  2  0 2X 2X 2X  0  0
 0  0 2X  0  0 2X 2X 2X 2X 2X 2X 2X  0  0  0  0  0  0  0  0 2X 2X 2X 2X 2X 2X 2X 2X  0  0  0  0  0  0 2X 2X 2X 2X 2X 2X  0  0  0  0 2X 2X  0  0  0  0 2X 2X 2X 2X 2X 2X 2X 2X  0  0  0  0  0  0 2X 2X  0  0 2X 2X  0  0 2X 2X  0 2X 2X 2X 2X  0 2X 2X  0  0 2X 2X  0  0 2X 2X  0  0  0  0  0  0  0 2X
 0  0  0 2X 2X 2X 2X  0  0  0 2X 2X 2X 2X  0  0  0  0 2X 2X 2X 2X  0  0  0  0 2X 2X 2X 2X  0  0  0 2X 2X  0  0 2X 2X 2X 2X  0 2X 2X  0  0  0  0 2X 2X 2X 2X  0  0  0  0 2X 2X 2X 2X  0  0  0 2X 2X  0 2X  0  0 2X  0 2X 2X  0 2X 2X  0  0 2X 2X  0 2X 2X  0  0 2X 2X  0  0 2X 2X  0  0  0  0  0 2X 2X

generates a code of length 98 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 97.

Homogenous weight enumerator: w(x)=1x^0+56x^97+168x^98+12x^100+8x^102+1x^104+8x^105+1x^112+1x^120

The gray image is a code over GF(2) with n=784, k=8 and d=388.
This code was found by Heurico 1.16 in 1.81 seconds.