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

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

Homogenous weight enumerator: w(x)=1x^0+180x^76+12x^78+48x^80+12x^82+2x^84+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 3.55 seconds.