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

generates a code of length 78 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 74.

Homogenous weight enumerator: w(x)=1x^0+4x^74+86x^75+122x^76+120x^77+314x^78+212x^79+68x^80+88x^81+6x^83+1x^88+1x^90+1x^130

The gray image is a code over GF(2) with n=624, k=10 and d=296.
This code was found by Heurico 1.16 in 0.5 seconds.