The generator matrix

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

generates a code of length 86 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 79.

Homogenous weight enumerator: w(x)=1x^0+162x^79+872x^80+1546x^81+2470x^82+3042x^83+3272x^84+3506x^85+3853x^86+3596x^87+3190x^88+2466x^89+1808x^90+1238x^91+822x^92+462x^93+231x^94+70x^95+78x^96+36x^97+20x^98+4x^99+20x^100+2x^102+1x^104

The gray image is a code over GF(2) with n=688, k=15 and d=316.
This code was found by Heurico 1.16 in 13.9 seconds.