The generator matrix

1 0 0 0 0 0 0 1 1 1 2 0 0 2 1 1 2 1 0 0 1 0 1 1 1 1 0 1 0 1 1 2 2 2 0 1 1 0 1 1 2 2 0 1 2 0 0 1 1 1 2 0 1 0 1 2 1 1 1 2 1 2 0 1 0 1 2 1 2
0 1 0 0 0 0 0 0 0 0 0 1 2 1 1 1 1 3 1 0 3 1 3 2 3 2 1 2 1 3 2 2 0 0 1 2 1 1 2 2 0 1 2 1 1 1 1 3 2 1 1 0 2 0 3 1 0 3 0 2 1 1 0 2 2 1 1 3 1
0 0 1 0 0 0 0 0 0 0 0 0 1 3 1 1 2 3 3 1 1 2 0 1 0 3 1 3 2 2 1 1 1 1 2 0 0 3 1 3 1 1 2 3 1 0 3 3 2 3 0 1 2 2 2 3 3 3 0 1 1 1 1 0 2 3 3 0 0
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 0 1 1 1 3 1 3 3 1 1 1 2 3 0 3 1 3 1 3 0 2 0 1 0 1 2 2 3 3 2 3 0 3 0 3 0 1 1 3 0 1 3
0 0 0 0 1 0 0 0 1 1 1 2 0 1 0 2 0 3 1 2 1 1 3 3 0 0 0 2 3 2 3 3 2 1 0 3 1 0 2 2 3 2 1 3 1 3 3 3 0 0 3 3 0 1 1 2 1 1 0 0 1 3 1 1 1 0 3 0 3
0 0 0 0 0 1 0 1 0 1 3 2 0 3 0 3 1 0 0 1 3 3 3 2 1 3 2 2 0 0 3 3 2 2 3 0 1 3 0 1 3 2 1 2 0 3 1 0 1 0 2 1 3 0 1 3 2 3 0 3 3 1 0 1 0 1 1 1 1
0 0 0 0 0 0 1 1 3 2 1 1 1 2 3 0 1 2 1 2 1 0 3 3 2 1 2 2 1 1 2 3 3 0 3 3 1 1 1 0 0 1 0 1 2 3 2 0 0 2 0 0 0 1 0 0 0 0 0 2 0 3 0 0 1 2 1 0 2
0 0 0 0 0 0 0 2 2 0 2 2 2 0 2 0 2 0 2 0 2 0 2 2 0 2 0 0 2 2 0 2 2 0 2 2 2 2 0 2 2 0 2 0 2 0 2 2 2 2 2 2 2 0 2 2 2 2 2 0 2 2 2 2 2 2 0 2 0

generates a code of length 69 over Z4 who´s minimum homogenous weight is 55.

Homogenous weight enumerator: w(x)=1x^0+104x^55+236x^56+336x^57+493x^58+660x^59+833x^60+992x^61+1200x^62+1288x^63+1507x^64+1636x^65+1780x^66+2040x^67+2048x^68+2088x^69+2180x^70+2162x^71+1889x^72+1788x^73+1606x^74+1416x^75+1175x^76+888x^77+738x^78+520x^79+415x^80+284x^81+158x^82+124x^83+80x^84+48x^85+34x^86+6x^87+7x^88+4x^89+3x^90+1x^112

The gray image is a code over GF(2) with n=138, k=15 and d=55.
This code was found by Heurico 1.10 in 57.4 seconds.