The generator matrix

 1  0  1  1  1 X^2+X  1  1 X+2  1  1 X^2+2  1  1 X^2+2  1  1 X+2  1  1 X^2+X  1  1  0  1  1  2  1  1 X^2+X+2  1  1  1  1 X^2  X  1  1  1  1  2 X^2+X+2  1  1  1  1 X^2  X  X  X  0  X  X X^2+2  X  X  0  1  1  X  X X^2+2  1  1  X X^2+2 X^2+2  X X^2  1  1 X^2  2  1  1  1  1  1
 0  1 X+1 X^2+X X^2+1  1 X^2+2 X^2+X+3  1 X+2  3  1  0 X+1  1 X^2+X X^2+1  1 X^2+2 X^2+X+3  1 X+2  3  1  2 X+3  1 X^2+X+2 X^2+3  1 X^2  X X^2+X+1  1  1  1  2 X^2+X+2 X+3 X^2+3  1  1 X^2  X X^2+X+1  1  1  1  0 X^2+X  X X^2+2 X+2  X  0 X^2+X  X X^2+2 X^2+2 X^2+2 X+2  X X^2+X+3 X^2+X+3 X^2  1  1  X  X X+2  3  X  X X^2+X X^2+1 X^2+X+2 X^2+2 X+2
 0  0  2  2  0  2  2  0  0  0  2  2  2  0  0  0  2  2  0  2  0  2  0  2  2  2  2  0  0  0  0  2  0  2  0  2  0  2  0  2  0  2  2  0  2  0  2  0  2  2  2  2  2  2  0  0  0  0  2  0  0  0  0  2  2  0  2  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+59x^76+160x^77+160x^78+50x^79+16x^80+32x^81+13x^82+10x^83+4x^84+4x^85+3x^86

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 0.297 seconds.