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  1
 0  X  2 3X+2  0 3X+2  2 3X  0 3X+2 3X  2  0 3X+2  2  X  0 3X+2  2 3X  0 3X+2  2  X  0 3X+2  2  X  0 X+2 2X+2  X 2X X+2 2X+2 3X 2X X+2 2X+2  X 2X X+2 2X+2  X 2X X+2 3X 2X+2 2X 3X+2  2 3X X+2 2X+2 3X 2X 2X X+2 2X+2 3X 2X X+2 2X+2  X  0 3X+2  2 3X 3X+2 3X+2  0
 0  0 2X  0  0  0 2X  0  0 2X 2X 2X  0 2X 2X 2X 2X  0  0 2X 2X 2X  0  0 2X  0  0 2X 2X 2X  0  0 2X 2X 2X 2X 2X 2X 2X 2X  0  0  0  0  0  0  0  0 2X 2X  0  0  0 2X  0 2X  0 2X 2X 2X  0  0  0 2X  0  0  0  0 2X 2X  0
 0  0  0 2X  0  0  0 2X 2X 2X 2X 2X 2X  0 2X  0 2X  0  0 2X 2X 2X  0  0  0 2X 2X  0  0  0 2X 2X  0  0  0  0 2X 2X 2X 2X 2X 2X 2X 2X  0  0  0  0  0  0 2X  0 2X 2X 2X 2X 2X 2X  0  0  0  0  0 2X  0  0  0 2X 2X  0  0
 0  0  0  0 2X 2X 2X 2X 2X  0 2X  0  0 2X 2X  0  0  0  0  0 2X 2X 2X 2X 2X 2X 2X 2X  0  0  0  0  0 2X 2X  0 2X 2X  0  0  0  0 2X 2X  0  0 2X 2X 2X  0  0  0 2X 2X  0  0 2X  0  0 2X 2X 2X  0 2X  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+18x^68+96x^69+109x^70+576x^71+109x^72+96x^73+17x^74+1x^78+1x^138

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 0.359 seconds.