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

generates a code of length 78 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 76.

Homogenous weight enumerator: w(x)=1x^0+71x^76+96x^77+198x^78+96x^79+32x^80+8x^82+5x^84+3x^88+2x^110

The gray image is a code over GF(2) with n=624, k=9 and d=304.
This code was found by Heurico 1.16 in 0.469 seconds.