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

generates a code of length 89 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 86.

Homogenous weight enumerator: w(x)=1x^0+200x^86+348x^88+304x^90+94x^92+72x^94+2x^96+2x^108+1x^128

The gray image is a code over GF(2) with n=712, k=10 and d=344.
This code was found by Heurico 1.16 in 11.3 seconds.