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

generates a code of length 87 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 86.

Homogenous weight enumerator: w(x)=1x^0+7x^86+106x^87+7x^88+4x^95+2x^103+1x^110

The gray image is a code over GF(2) with n=696, k=7 and d=344.
This code was found by Heurico 1.16 in 0.438 seconds.