The generator matrix

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

generates a code of length 87 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 80.

Homogenous weight enumerator: w(x)=1x^0+163x^80+956x^81+1503x^82+2378x^83+2870x^84+3548x^85+3405x^86+3974x^87+3526x^88+3242x^89+2233x^90+2124x^91+1228x^92+706x^93+401x^94+250x^95+102x^96+70x^97+33x^98+10x^99+22x^100+22x^101+1x^106

The gray image is a code over GF(2) with n=696, k=15 and d=320.
This code was found by Heurico 1.16 in 15.5 seconds.