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  1  1  1  1  1  1  1  1  X  X  X  X  X  X  X  1  1  1 2X+2 2X+2 2X+2 2X+2  1 2X+2 2X+2 2X+2  1  1  X  1  1  X  X  X  X  X  X  X  X  1  1  1  1  1  1  1  1  X  1  1  1  1  1  1  1  1 2X+2  1
 0 2X  0  0  0 2X 2X 2X  0  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 2X 2X  0 2X 2X  0  0  0  0 2X 2X 2X 2X  0  0 2X 2X  0 2X 2X  0  0  0 2X 2X 2X 2X  0  0  0  0 2X 2X  0 2X 2X  0  0 2X 2X  0 2X 2X  0  0  0  0 2X  0 2X 2X 2X  0  0  0  0 2X  0 2X 2X 2X  0  0
 0  0 2X  0 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 2X 2X  0  0  0 2X 2X 2X 2X  0  0  0 2X 2X  0 2X 2X  0  0  0 2X 2X 2X  0  0 2X  0 2X 2X 2X 2X  0  0  0  0 2X 2X  0 2X 2X  0  0  0  0 2X 2X 2X 2X  0  0  0  0  0 2X 2X 2X 2X  0  0  0  0
 0  0  0 2X 2X  0 2X 2X  0 2X 2X  0  0 2X 2X  0  0 2X 2X  0  0 2X 2X  0  0 2X 2X  0  0 2X 2X  0 2X 2X  0  0  0 2X 2X  0 2X 2X  0  0 2X 2X  0 2X 2X  0  0  0 2X 2X  0 2X 2X  0 2X 2X  0  0 2X 2X  0  0 2X  0 2X  0 2X 2X  0  0  0 2X 2X  0  0 2X 2X  0  0 2X 2X  0  0  0 2X 2X  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+7x^96+106x^97+7x^98+4x^105+2x^113+1x^130

The gray image is a code over GF(2) with n=776, k=7 and d=384.
This code was found by Heurico 1.16 in 0.703 seconds.