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  X  0  1  X  2  1  1  1  1  1  1  1  X  0  X  2  X  X  X  X  X 2X  X 2X+2  X 2X  X 2X+2  X  X  X  X  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X
 0  X  2 3X+2  0 3X+2  2 3X 2X X+2 2X+2 3X 2X X+2 2X+2  X  0 3X+2  2 3X  0 3X+2  2 3X 3X+2  X 2X 3X  X X+2 2X+2  X 2X X+2 2X+2  X 3X+2  X 3X  X  0  2 2X 2X+2 X+2  X  X  X X+2  X  X  X  0  2 2X 2X+2  0  2  0  2 2X 2X+2 2X 2X+2 3X+2 3X 3X+2 3X X+2  X X+2  X  0  2  0 2X  2
 0  0 2X 2X 2X  0  0 2X 2X 2X  0  0  0  0 2X 2X  0  0 2X 2X 2X 2X  0  0  0 2X 2X 2X  0 2X  0  0  0  0 2X 2X 2X  0  0 2X 2X 2X 2X 2X 2X 2X  0  0  0  0 2X 2X  0  0  0  0  0 2X 2X  0 2X  0  0 2X  0 2X 2X  0 2X  0  0 2X  0 2X 2X  0  0

generates a code of length 77 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 76.

Homogenous weight enumerator: w(x)=1x^0+48x^76+160x^77+36x^78+5x^80+4x^86+1x^88+1x^104

The gray image is a code over GF(2) with n=616, k=8 and d=304.
This code was found by Heurico 1.16 in 0.453 seconds.