The generator matrix

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

generates a code of length 30 over Z4 who´s minimum homogenous weight is 20.

Homogenous weight enumerator: w(x)=1x^0+85x^20+98x^21+339x^22+386x^23+651x^24+732x^25+930x^26+1172x^27+1252x^28+1690x^29+1549x^30+1652x^31+1398x^32+1322x^33+1019x^34+732x^35+610x^36+292x^37+236x^38+82x^39+90x^40+26x^41+23x^42+8x^43+9x^44

The gray image is a code over GF(2) with n=60, k=14 and d=20.
This code was found by Heurico 1.16 in 21.6 seconds.