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

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

Homogenous weight enumerator: w(x)=1x^0+7x^88+106x^89+7x^90+4x^97+2x^105+1x^114

The gray image is a code over GF(2) with n=712, k=7 and d=352.
This code was found by Heurico 1.16 in 0.5 seconds.