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

generates a code of length 89 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 88.

Homogenous weight enumerator: w(x)=1x^0+14x^88+220x^89+14x^90+4x^97+1x^98+1x^112+1x^114

The gray image is a code over GF(2) with n=712, k=8 and d=352.
This code was found by Heurico 1.16 in 0.797 seconds.