The generator matrix

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

generates a code of length 89 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 84.

Homogenous weight enumerator: w(x)=1x^0+189x^84+688x^85+1081x^86+1188x^87+955x^88+1014x^89+781x^90+690x^91+370x^92+398x^93+306x^94+186x^95+120x^96+104x^97+85x^98+16x^99+12x^100+4x^101+1x^102+1x^108+2x^110

The gray image is a code over GF(2) with n=712, k=13 and d=336.
This code was found by Heurico 1.16 in 1.44 seconds.