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 X 1 X 1 1 0 1 1 1 1 1 X^2 X X 1 X 1 0 1 1 1 0 1 X 1 X X X 1 1 0 1 1 1 0 X X 0 1 1 X X X^2 X 1 0 X 0 0 0 0 0 0 X^2 X X^2+X X^2+X X^2+X X^2+X X^2+X X^2+X X^2 0 X^2 X^2+X X X X^2 X X^2+X X^2+X 0 X^2 0 X 0 X^2+X X^2+X X^2+X X^2+X 0 X^2 X^2 0 X X^2 X^2 0 X X^2 X^2+X X^2 0 X X^2 X X^2+X 0 X^2+X X X^2+X 0 X^2+X X X^2 X^2+X X^2 X^2 X X^2+X X^2 X^2+X X^2 X^2+X X^2+X 0 X^2 X X X^2+X X 0 0 0 0 X 0 0 0 X X^2+X X^2+X X X X^2 X^2+X X 0 X^2 0 X^2+X X^2+X 0 X X X 0 X^2 X X^2 X^2 X^2+X X^2+X 0 0 X^2 X X^2+X 0 X X^2 X^2 X X^2 X^2+X X X 0 0 X^2+X X X^2 X^2+X 0 X^2 X^2+X X^2 0 X^2 0 X^2 X^2 X^2+X X^2 X^2 0 X^2+X X^2 X X X X^2 X^2+X X^2 X^2+X 0 X^2 X^2 X^2+X X^2 0 0 0 0 X 0 X X X 0 X^2 0 X X^2+X X X X^2+X X^2 0 0 X^2 X^2+X X X X^2 0 X^2 X^2+X X^2 X 0 X^2+X X X^2+X X^2+X X^2 X^2+X X X X 0 0 X 0 X^2+X X 0 X^2 X^2+X X^2 X X X^2+X X^2+X 0 X X^2+X X^2+X X^2 0 X^2+X X X^2+X X^2+X X X 0 X X^2 0 X X X^2+X X^2+X X X^2 X X^2 0 0 0 0 0 X X X^2 X^2+X X X^2 X 0 X X^2 X^2+X X^2 0 0 X X^2+X X 0 X^2 X^2+X X^2 X 0 X X 0 X X^2+X X^2+X 0 X^2 0 X X X^2 0 X^2+X X^2 X^2+X X^2+X X^2+X X^2 X^2+X X X^2+X X X^2 X^2+X X^2+X X 0 0 X^2+X 0 X^2 X X^2+X 0 X^2 X X^2 0 X^2 X^2 0 X^2+X X^2+X X^2+X X^2 0 X X^2+X X^2+X X^2+X 0 0 0 0 0 X^2 X^2 X^2 0 0 0 X^2 X^2 0 0 0 X^2 X^2 X^2 X^2 0 X^2 0 0 X^2 X^2 X^2 X^2 0 X^2 0 X^2 0 X^2 0 X^2 X^2 X^2 X^2 X^2 X^2 X^2 0 X^2 X^2 X^2 0 0 X^2 0 X^2 X^2 X^2 X^2 X^2 0 0 X^2 0 0 X^2 X^2 0 X^2 X^2 X^2 X^2 X^2 X^2 X^2 0 X^2 X^2 0 0 0 X^2 0 generates a code of length 78 over Z2[X]/(X^3) who´s minimum homogenous weight is 69. Homogenous weight enumerator: w(x)=1x^0+60x^69+109x^70+162x^71+197x^72+246x^73+262x^74+246x^75+321x^76+338x^77+366x^78+378x^79+302x^80+250x^81+189x^82+146x^83+144x^84+84x^85+72x^86+58x^87+39x^88+36x^89+21x^90+28x^91+19x^92+10x^93+5x^94+6x^95+1x^112 The gray image is a linear code over GF(2) with n=312, k=12 and d=138. This code was found by Heurico 1.16 in 1.81 seconds.