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

generates a code of length 97 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 96.

Homogenous weight enumerator: w(x)=1x^0+14x^96+216x^97+15x^98+8x^105+1x^112+1x^114

The gray image is a code over GF(2) with n=776, k=8 and d=384.
This code was found by Heurico 1.16 in 1.13 seconds.