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

generates a code of length 88 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 84.

Homogenous weight enumerator: w(x)=1x^0+186x^84+156x^86+384x^87+606x^88+384x^89+144x^90+177x^92+4x^94+5x^96+1x^172

The gray image is a code over GF(2) with n=704, k=11 and d=336.
This code was found by Heurico 1.16 in 8.72 seconds.