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

generates a code of length 71 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+45x^68+32x^69+72x^70+704x^71+106x^72+32x^73+24x^74+7x^76+1x^136

The gray image is a code over GF(2) with n=568, k=10 and d=272.
This code was found by Heurico 1.16 in 0.375 seconds.