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

generates a code of length 87 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 82.

Homogenous weight enumerator: w(x)=1x^0+160x^82+782x^83+877x^84+1260x^85+1100x^86+1046x^87+573x^88+732x^89+469x^90+450x^91+191x^92+188x^93+144x^94+122x^95+52x^96+28x^97+13x^98+1x^100+2x^102+1x^108

The gray image is a code over GF(2) with n=696, k=13 and d=328.
This code was found by Heurico 1.16 in 1.22 seconds.