The generator matrix

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

generates a code of length 86 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 81.

Homogenous weight enumerator: w(x)=1x^0+146x^81+917x^82+708x^83+1456x^84+754x^85+1108x^86+644x^87+860x^88+434x^89+439x^90+128x^91+317x^92+94x^93+121x^94+20x^95+21x^96+12x^97+5x^98+4x^99+1x^100+1x^102+1x^106

The gray image is a code over GF(2) with n=688, k=13 and d=324.
This code was found by Heurico 1.16 in 1.2 seconds.