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

generates a code of length 86 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 82.

Homogenous weight enumerator: w(x)=1x^0+66x^82+16x^83+218x^84+112x^85+242x^86+112x^87+164x^88+16x^89+38x^90+28x^92+6x^94+2x^96+2x^100+1x^128

The gray image is a code over GF(2) with n=688, k=10 and d=328.
This code was found by Heurico 1.16 in 0.735 seconds.