The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+256x^76+96x^78+106x^80+40x^84+12x^88+1x^96

The gray image is a code over GF(2) with n=312, k=9 and d=152.
This code was found by Heurico 1.16 in 4.64 seconds.