The generator matrix

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

generates a code of length 77 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 70.

Homogenous weight enumerator: w(x)=1x^0+322x^70+1600x^71+2838x^72+4322x^73+5673x^74+6754x^75+7630x^76+7990x^77+7723x^78+6638x^79+5080x^80+4078x^81+2448x^82+1266x^83+644x^84+298x^85+125x^86+58x^87+15x^88+16x^89+13x^90+4x^91

The gray image is a code over GF(2) with n=616, k=16 and d=280.
This code was found by Heurico 1.16 in 43.3 seconds.