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

generates a code of length 70 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+9x^68+64x^69+109x^70+60x^71+6x^72+2x^73+2x^74+2x^85+1x^106

The gray image is a code over GF(2) with n=560, k=8 and d=272.
This code was found by Heurico 1.16 in 0.235 seconds.