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

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

Homogenous weight enumerator: w(x)=1x^0+24x^78+7x^80

The gray image is a code over GF(2) with n=608, k=5 and d=312.
This code was found by Heurico 1.16 in 0.282 seconds.