The generator matrix

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

generates a code of length 79 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 77.

Homogenous weight enumerator: w(x)=1x^0+160x^77+222x^78+260x^79+220x^80+152x^81+2x^82+4x^83+1x^92+1x^96+1x^124

The gray image is a code over GF(2) with n=632, k=10 and d=308.
This code was found by Heurico 1.16 in 0.391 seconds.