The generator matrix

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

generates a code of length 68 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 61.

Homogenous weight enumerator: w(x)=1x^0+264x^61+1358x^62+2692x^63+3783x^64+5456x^65+7272x^66+7808x^67+8641x^68+8068x^69+6842x^70+5540x^71+3643x^72+2116x^73+1216x^74+464x^75+199x^76+92x^77+54x^78+8x^79+5x^80+4x^81+8x^82+2x^86

The gray image is a code over GF(2) with n=544, k=16 and d=244.
This code was found by Heurico 1.16 in 39.3 seconds.