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 2X  1  1  1  1  1  1  1  1 2X  X  1  X  X  X
 0  X  0 X+2  2 3X+2 2X+2  X  2 X+2 2X 3X 2X X+2  2  X 2X X+2  2 3X  0  X 2X+2 3X+2  0 3X+2  2  X  0 X+2  2  X  0 X+2  2  X 2X+2 3X 2X 3X+2 2X 3X+2 2X+2 3X  2  X 3X+2 2X 2X 3X 2X+2  X  X  2 2X+2 3X+2 3X+2 2X  2 3X  0 2X 2X 2X X+2 3X+2 3X+2  0 X+2  0 2X 2X  X 3X+2 2X+2 X+2  2 3X+2
 0  0 2X+2  0  2  2  0  2 2X+2  0  2  0  0 2X+2  0 2X+2  2 2X  2 2X 2X  2 2X  2 2X 2X 2X 2X 2X+2 2X+2 2X+2 2X+2  0  0  2  2 2X+2 2X+2  2  2  0  0 2X  0 2X  0 2X+2 2X+2 2X+2 2X  0 2X  2  0 2X+2  2 2X+2  2  2 2X+2 2X 2X 2X  0 2X 2X  0 2X+2 2X+2  0  0 2X+2 2X+2 2X+2 2X+2  0  2  2
 0  0  0 2X  0  0 2X 2X 2X 2X 2X  0 2X  0  0 2X  0 2X 2X  0 2X 2X  0  0  0 2X 2X  0 2X  0  0 2X  0  0  0  0 2X  0  0 2X 2X  0  0 2X 2X 2X 2X  0 2X 2X  0 2X  0 2X  0 2X 2X 2X 2X  0 2X 2X  0  0  0  0 2X  0 2X 2X  0 2X 2X  0 2X 2X 2X  0
 0  0  0  0 2X  0  0  0  0 2X 2X 2X 2X 2X 2X 2X  0  0  0  0 2X 2X 2X 2X  0 2X  0 2X 2X  0 2X  0 2X  0  0 2X 2X  0 2X  0  0 2X  0 2X 2X  0  0 2X  0 2X  0  0  0 2X  0 2X 2X  0 2X 2X  0 2X 2X  0 2X  0 2X 2X 2X  0 2X 2X 2X  0  0 2X  0 2X

generates a code of length 78 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 74.

Homogenous weight enumerator: w(x)=1x^0+204x^74+32x^75+402x^76+192x^77+464x^78+288x^79+214x^80+156x^82+86x^84+8x^86+1x^144

The gray image is a code over GF(2) with n=624, k=11 and d=296.
This code was found by Heurico 1.16 in 18.1 seconds.