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  0  X  2  1  X 2X  1  1  X 2X+2  1  1  1  1  1  X  X  X  X  X  0  X  2  1  1  1  X 2X  1  X 2X+2  1  1  1  1  X  X  X  X 2X+2  0 2X+2 2X  X  0  X  2  1  1  X  X 2X 2X+2  1  1 2X+2 2X+2  1  1  1  1  1
 0  X  2 3X+2 2X X+2 2X+2 3X  0 3X+2  2 3X 2X X+2 2X+2  X  0 3X+2  2 3X 2X X+2 2X+2  X 3X+2  X 3X  X  0 X+2  X 3X+2  2  X  X 3X 2X X+2 2X+2  X  0  2 2X 2X+2 3X+2  X 3X  X  0  2 3X+2 X+2  X 3X  X  X 2X 2X+2 X+2  X  0  2 2X 2X+2  2 2X+2 2X+2 2X+2 3X+2  X 3X  X  0 2X X+2  X  X  X  2 2X+2  0 2X  0 2X  2 2X+2  0

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+4x^86+88x^87+7x^88+24x^89+2x^90+1x^98+1x^106

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 0.453 seconds.