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 0 1 X^2 1 X^2 1 1 1 1 1 1 1 X 1 X^2 1 1 0 1 1 0 1 1 1 1 X 1 0 1 1 1 1 1 X^2 X^2 1 X X X 1 X 1 1
0 X 0 0 0 0 0 X^2 X^2 X X^2+X X X X X^2+X X 0 X^2+X X^2 X X^2 X 0 X X^2 X^2+X X 0 X^2+X X^2 X^2+X 0 0 X^2 0 X X X^2+X X^2 X^2 X^2+X 0 0 X^2 X^2+X X X 0 X X^2 0 X X^2+X X^2+X 0 X X^2+X X^2 X^2 0 0 X^2+X X^2 0 X X^2+X X^2 X^2 X^2 X^2+X X^2 X^2+X 0 0 0 X^2+X X X 0 X^2 X X^2 X^2 X X 0
0 0 X 0 0 X^2 X^2+X X X X X X X^2+X 0 0 0 X^2 X^2 X^2+X X X^2 0 0 X^2+X X^2 X^2 X X^2+X 0 X X X^2+X X X^2+X X^2+X 0 X^2 0 0 X^2 X^2+X X^2 X X X X^2 X 0 X^2 0 0 X^2+X 0 X X^2 X^2 X^2 X X^2+X X 0 X^2+X X X X^2 X X^2+X 0 X^2+X 0 X^2 X^2+X 0 X X X X^2 X X^2+X 0 X X^2 X X^2+X X^2 0
0 0 0 X 0 X^2+X X^2+X X X^2 X^2+X X^2+X 0 0 X X X^2 X X^2 X^2+X 0 X^2+X 0 X^2 X X^2 X X X^2 X X 0 0 X^2 X 0 X^2+X X^2 X^2+X X X^2 0 0 X 0 X^2+X X^2 X^2+X X X X X^2 0 X^2 X^2 0 X^2 0 0 X^2 0 X^2 X^2+X X X X^2 X^2+X 0 X^2+X 0 X X X X 0 X X X X^2 X 0 X^2 X^2+X 0 0 X^2 0
0 0 0 0 X X X^2 X^2+X X X^2+X X^2 X^2 X X^2 X^2+X X X X^2 X^2 X^2+X 0 X^2+X 0 X^2+X X^2+X X^2+X 0 X^2+X 0 X 0 X^2 0 X^2+X X 0 X^2 X^2+X X^2+X 0 X^2+X X^2+X 0 X^2 X^2+X X^2+X 0 X^2+X 0 0 X^2 X^2 X^2+X X^2 X X X^2 X 0 X^2 X 0 0 X^2+X X^2+X X X X^2+X X X X^2+X X^2+X X^2+X X X^2+X 0 0 X^2 0 X X^2+X 0 X^2 X^2+X X 0
generates a code of length 86 over Z2[X]/(X^3) who´s minimum homogenous weight is 78.
Homogenous weight enumerator: w(x)=1x^0+32x^78+72x^79+102x^80+138x^81+157x^82+154x^83+141x^84+174x^85+187x^86+192x^87+193x^88+140x^89+79x^90+60x^91+47x^92+38x^93+40x^94+18x^95+20x^96+14x^97+16x^98+14x^99+8x^100+8x^101+2x^103+1x^142
The gray image is a linear code over GF(2) with n=344, k=11 and d=156.
This code was found by Heurico 1.16 in 0.732 seconds.