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

generates a code of length 68 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 65.

Homogenous weight enumerator: w(x)=1x^0+12x^65+51x^66+116x^67+662x^68+116x^69+52x^70+12x^71+1x^72+1x^130

The gray image is a code over GF(2) with n=544, k=10 and d=260.
This code was found by Heurico 1.16 in 75.3 seconds.