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

generates a code of length 78 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 72.

Homogenous weight enumerator: w(x)=1x^0+57x^72+97x^74+213x^76+256x^77+847x^78+256x^79+192x^80+49x^82+38x^84+29x^86+10x^88+2x^90+1x^148

The gray image is a code over GF(2) with n=624, k=11 and d=288.
This code was found by Heurico 1.16 in 0.625 seconds.