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 X 0 X X^2+2 X 2 1 1 X X^2 1 1 1 1 1 1 X X X X X 0 X X^2+2 1 1 X 2 1 X X^2 1 1 1 1 1 X X X X X^2 0 X^2 2 X 0 X X^2+2 X X 2 X^2 1 1 1 1 1 1 1 1 1 1
0 X X^2+2 X^2+X 2 X^2+X+2 X^2 X+2 0 X^2+X X^2+2 X+2 2 X^2+X+2 X^2 X 0 X^2+X X^2+2 X+2 2 X^2+X+2 X^2 X X^2+X X X+2 X X^2+X+2 X 0 X^2+X X X X^2+2 X+2 2 X^2+X+2 X^2 X 0 X^2+2 2 X^2 X^2+X X X+2 X 0 X^2+2 X^2+X+2 X 2 X X X^2 X^2+X X+2 X^2+X+2 X 0 X^2+2 2 X^2 X^2+2 X^2 X^2 X^2 X^2+X X X+2 X X^2+X+2 X X X 0 2 X^2+2 X^2 0 2 X^2+2 X^2 0 2
generates a code of length 86 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 86.
Homogenous weight enumerator: w(x)=1x^0+112x^86+3x^88+8x^90+2x^92+1x^100+1x^108
The gray image is a code over GF(2) with n=688, k=7 and d=344.
This code was found by Heurico 1.16 in 0.438 seconds.