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  X  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 2X+2
 0  2  0 2X+2  0  0 2X+2  2  0  0 2X+2  2  0  0 2X+2  2  0  0 2X+2  2  0  0 2X+2  2  0  0 2X+2  2 2X  2 2X 2X 2X 2X+2 2X  2 2X+2  2 2X+2 2X 2X  2 2X+2 2X 2X  2 2X+2 2X 2X  0  0  2 2X+2 2X 2X  2 2X+2 2X 2X  2 2X+2 2X 2X  2 2X+2  0  0 2X+2  2  0  0 2X+2  2  0 2X 2X+2 2X+2 2X  2  0  2  2 2X+2  2 2X+2 2X+2 2X+2
 0  0  2 2X+2  0  2 2X+2  0  0  2 2X+2  0  0  2 2X+2  0 2X 2X+2  2 2X 2X 2X+2  2 2X 2X 2X+2  2 2X 2X 2X 2X+2  2  0  2 2X+2  2 2X  2 2X 2X 2X+2  2 2X 2X 2X+2  2 2X 2X 2X+2 2X 2X+2 2X+2  0  0  2 2X+2  0  0  2 2X+2  0  0  2 2X+2  0  0  2 2X+2  0  0  2  2 2X  0 2X+2 2X+2 2X 2X 2X+2 2X  0 2X  0 2X+2  0  0 2X+2
 0  0  0 2X  0  0 2X  0 2X 2X  0 2X 2X 2X  0 2X 2X 2X 2X 2X 2X 2X 2X 2X  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 2X 2X 2X 2X 2X 2X 2X 2X  0  0 2X 2X 2X 2X  0  0 2X 2X 2X 2X  0  0  0  0  0  0  0  0  0  0  0  0 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X  0 2X  0 2X
 0  0  0  0 2X 2X 2X 2X 2X 2X  0 2X  0  0 2X  0  0  0  0  0 2X 2X 2X 2X 2X 2X 2X 2X  0  0  0  0  0  0  0  0  0 2X 2X 2X 2X 2X 2X 2X 2X  0  0  0  0  0  0  0  0  0  0  0  0 2X 2X 2X 2X 2X 2X 2X 2X  0  0  0  0 2X 2X 2X 2X 2X  0 2X  0  0  0 2X 2X 2X  0  0 2X 2X 2X

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

Homogenous weight enumerator: w(x)=1x^0+121x^84+224x^86+256x^87+378x^88+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 96.8 seconds.