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

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

Homogenous weight enumerator: w(x)=1x^0+118x^76+1x^80+6x^84+2x^88

The gray image is a code over GF(2) with n=608, k=7 and d=304.
This code was found by Heurico 1.16 in 0.547 seconds.