The generator matrix

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

generates a code of length 62 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 55.

Homogenous weight enumerator: w(x)=1x^0+752x^55+2160x^56+3988x^57+7245x^58+10400x^59+14392x^60+17384x^61+18514x^62+17366x^63+14669x^64+10748x^65+6925x^66+3472x^67+1790x^68+748x^69+286x^70+134x^71+42x^72+26x^73+21x^74+4x^75+2x^76+2x^81+1x^82

The gray image is a code over GF(2) with n=496, k=17 and d=220.
This code was found by Heurico 1.16 in 292 seconds.