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

generates a code of length 88 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 82.

Homogenous weight enumerator: w(x)=1x^0+68x^82+268x^83+250x^84+464x^85+391x^86+460x^87+452x^88+450x^89+306x^90+404x^91+228x^92+192x^93+70x^94+40x^95+4x^96+6x^97+8x^98+12x^99+8x^100+8x^101+4x^102+1x^104+1x^142

The gray image is a code over GF(2) with n=704, k=12 and d=328.
This code was found by Heurico 1.16 in 1.17 seconds.