The generator matrix

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

generates a code of length 86 over Z4[X]/(X^2+2,2X) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+81x^78+384x^79+304x^80+718x^81+506x^82+902x^83+476x^84+766x^85+529x^86+708x^87+402x^88+584x^89+339x^90+494x^91+211x^92+278x^93+122x^94+168x^95+68x^96+86x^97+17x^98+32x^99+9x^100+4x^102+1x^104+2x^106

The gray image is a code over GF(2) with n=344, k=13 and d=156.
This code was found by Heurico 1.16 in 4.08 seconds.