The generator matrix

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

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+537x^68+608x^70+404x^72+344x^74+140x^76+8x^78+3x^80+2x^84+1x^116

The gray image is a code over GF(2) with n=568, k=11 and d=272.
This code was found by Heurico 1.16 in 76 seconds.