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

generates a code of length 77 over Z4[X]/(X^2+2,2X) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+3x^78+22x^79+3x^80+2x^83+1x^86

The gray image is a code over GF(2) with n=308, k=5 and d=156.
This code was found by Heurico 1.16 in 0.217 seconds.