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

generates a code of length 88 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 84.

Homogenous weight enumerator: w(x)=1x^0+186x^84+156x^86+384x^87+606x^88+384x^89+144x^90+177x^92+4x^94+5x^96+1x^172

The gray image is a code over GF(2) with n=704, k=11 and d=336.
This code was found by Heurico 1.16 in 8.58 seconds.