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 0 X 0 X 0 X 0 X X X X X X X^2 X X X^2 X X^2 X X^2 1 1 1 1 1 1 1 1 X 0 X 0 X 0 X X^2 1 1 X 1 1 1 1 1 1 X X X^2 X X 0 X X X X
0 X 0 X^2+X 0 X^2+X 0 X 0 X^2+X 0 X 0 X^2+X 0 X X^2 X^2+X X^2 X X^2 X^2+X X^2 X X^2 X^2+X X^2 X X^2 X^2+X X^2 X X^2+X X X^2+X X X^2+X X X^2+X X 0 0 X^2 X^2 0 X X X^2 X X X X X X 0 0 X^2 X^2 0 0 X^2 X^2 X^2+X X X^2+X X X^2+X X X X 0 0 X^2 X^2 X^2 0 0 X^2 X^2 0 X^2+X X 0 X X 0 0 X^2 0
0 0 X^2 0 0 0 X^2 0 0 X^2 0 X^2 X^2 X^2 X^2 X^2 X^2 0 X^2 0 X^2 0 X^2 0 0 X^2 0 X^2 0 X^2 0 X^2 0 0 0 0 X^2 X^2 X^2 X^2 0 X^2 X^2 0 X^2 X^2 X^2 X^2 X^2 X^2 0 0 0 0 0 0 0 0 0 X^2 X^2 0 0 0 X^2 X^2 0 0 X^2 X^2 0 X^2 X^2 X^2 0 X^2 X^2 X^2 X^2 0 X^2 X^2 X^2 X^2 X^2 X^2 0 X^2 0
0 0 0 X^2 0 0 0 X^2 X^2 X^2 X^2 X^2 X^2 0 X^2 0 0 0 0 0 X^2 X^2 X^2 X^2 X^2 X^2 X^2 X^2 0 0 0 0 0 0 X^2 X^2 X^2 X^2 0 0 X^2 X^2 X^2 X^2 0 0 0 0 X^2 X^2 X^2 X^2 0 0 0 0 0 0 X^2 X^2 X^2 X^2 0 0 0 0 X^2 X^2 X^2 X^2 X^2 X^2 0 X^2 X^2 0 0 0 0 0 X^2 0 X^2 0 X^2 0 0 X^2 0
0 0 0 0 X^2 X^2 X^2 X^2 X^2 0 0 X^2 0 X^2 X^2 0 0 0 X^2 X^2 X^2 X^2 0 0 0 X^2 X^2 0 X^2 0 0 X^2 0 X^2 X^2 0 0 X^2 X^2 0 X^2 0 0 X^2 X^2 X^2 0 X^2 0 X^2 X^2 0 0 X^2 0 X^2 X^2 0 X^2 X^2 X^2 X^2 X^2 0 0 X^2 0 X^2 X^2 0 0 0 0 0 0 X^2 0 0 X^2 X^2 X^2 X^2 X^2 0 0 0 0 X^2 0
generates a code of length 89 over Z2[X]/(X^3) who´s minimum homogenous weight is 86.
Homogenous weight enumerator: w(x)=1x^0+28x^86+32x^87+39x^88+64x^89+32x^90+32x^91+16x^92+7x^96+4x^102+1x^120
The gray image is a linear code over GF(2) with n=356, k=8 and d=172.
This code was found by Heurico 1.16 in 0.52 seconds.