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

generates a code of length 78 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 76.

Homogenous weight enumerator: w(x)=1x^0+198x^76+128x^77+384x^78+128x^79+175x^80+9x^84+1x^148

The gray image is a code over GF(2) with n=624, k=10 and d=304.
This code was found by Heurico 1.16 in 4.63 seconds.