The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+504x^62+1160x^63+2139x^64+2964x^65+3868x^66+3762x^67+4741x^68+3580x^69+3631x^70+2700x^71+1779x^72+888x^73+571x^74+226x^75+107x^76+68x^77+45x^78+8x^79+17x^80+4x^81+5x^82

The gray image is a code over GF(2) with n=544, k=15 and d=248.
This code was found by Heurico 1.16 in 109 seconds.