The generator matrix

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

generates a code of length 68 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 62.

Homogenous weight enumerator: w(x)=1x^0+96x^62+592x^63+1306x^64+1252x^65+2308x^66+1780x^67+2469x^68+1506x^69+2079x^70+1008x^71+966x^72+386x^73+288x^74+196x^75+73x^76+30x^77+11x^78+24x^79+1x^80+10x^81+2x^82

The gray image is a code over GF(2) with n=544, k=14 and d=248.
This code was found by Heurico 1.16 in 2.83 seconds.