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

generates a code of length 68 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 64.

Homogenous weight enumerator: w(x)=1x^0+250x^64+256x^66+512x^67+64x^68+512x^69+256x^70+168x^72+28x^80+1x^128

The gray image is a code over GF(2) with n=544, k=11 and d=256.
This code was found by Heurico 1.16 in 62.3 seconds.