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

generates a code of length 87 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 84.

Homogenous weight enumerator: w(x)=1x^0+113x^84+248x^86+256x^87+354x^88+8x^90+43x^92+1x^168

The gray image is a code over GF(2) with n=696, k=10 and d=336.
This code was found by Heurico 1.16 in 31.6 seconds.