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

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

Homogenous weight enumerator: w(x)=1x^0+248x^78+57x^80+176x^82+24x^86+3x^88+2x^96+1x^104

The gray image is a code over GF(2) with n=320, k=9 and d=156.
This code was found by Heurico 1.16 in 31.3 seconds.