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

generates a code of length 78 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+34x^78+4x^79+16x^80+4x^81+2x^82+1x^84+2x^86

The gray image is a code over GF(2) with n=624, k=6 and d=312.
This code was found by Heurico 1.16 in 0.36 seconds.