The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+226x^80+16x^82+476x^84+304x^86+531x^88+160x^90+192x^92+105x^96+28x^100+8x^104+1x^152

The gray image is a code over GF(2) with n=348, k=11 and d=160.
This code was found by Heurico 1.16 in 35.9 seconds.