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

generates a code of length 80 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+141x^78+244x^80+106x^82+8x^84+9x^86+1x^88+2x^108

The gray image is a code over GF(2) with n=640, k=9 and d=312.
This code was found by Heurico 1.16 in 1.33 seconds.