The generator matrix

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

generates a code of length 87 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 84.

Homogenous weight enumerator: w(x)=1x^0+30x^84+168x^85+292x^86+48x^87+282x^88+168x^89+26x^90+6x^92+1x^96+1x^106+1x^138

The gray image is a code over GF(2) with n=696, k=10 and d=336.
This code was found by Heurico 1.16 in 0.485 seconds.