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

generates a code of length 79 over Z4[X]/(X^2+2,2X) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+12x^78+88x^79+15x^80+8x^83+3x^86+1x^102

The gray image is a code over GF(2) with n=316, k=7 and d=156.
This code was found by Heurico 1.16 in 0.242 seconds.