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

generates a code of length 83 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+126x^78+54x^80+430x^82+1024x^83+176x^84+138x^86+24x^88+74x^90+1x^160

The gray image is a code over GF(2) with n=664, k=11 and d=312.
This code was found by Heurico 1.16 in 36.5 seconds.