The generator matrix

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

generates a code of length 87 over Z4[X]/(X^2+2,2X) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+130x^78+454x^79+666x^80+894x^81+874x^82+1098x^83+1262x^84+1172x^85+1281x^86+1142x^87+1268x^88+1214x^89+1070x^90+980x^91+811x^92+632x^93+471x^94+380x^95+257x^96+142x^97+72x^98+50x^99+21x^100+20x^101+2x^102+4x^103+2x^104+6x^105+2x^106+4x^107+2x^110

The gray image is a code over GF(2) with n=348, k=14 and d=156.
This code was found by Heurico 1.16 in 16.9 seconds.