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 X 1 1 1 1 1 1 1 1 1 X 1 0 1 1 1 X 1 1 1 1 X 1 1 1 1 1 X 1 X 1 1 1 X X^3 X 1 X^3 X^2 1 X X X^3+X^2 X^2 X^3+X^2 X 0 X 0 X 0 X^3 X^3+X X X^2 X^2+X X^2 X^3+X^2+X X^2 X^3+X^2 X^3+X^2+X X^3+X^2+X 0 X^3+X^2 X X^3+X^2+X X 0 X^3 X^3+X X^3+X^2+X X^2+X X^3 X^3+X^2+X X^2 X^2 X^3+X 0 X^2 X^2 X X^2+X X X^2 X^3+X^2+X 0 X^3+X^2+X 0 X X^3 X^3+X X^3+X^2 X^3+X^2+X 0 X^3+X^2 X^3+X^2+X X^2+X X^2 X^3 X X^3 X^2+X X^2+X X^3+X^2 X^3 X X^2 0 X^3+X X^3 X X^3+X^2 X^2 X^3+X X^3+X^2+X X^2 X^2+X X^3+X 0 X^2+X X^3+X^2 X^3 X^3 X X X^3+X^2 X X^3+X^2+X X X X X^3 0 0 X X X^3+X^2 X^3+X^2+X X^2+X X^2 X^2 X^3+X^2+X X 0 X^3 X^3+X^2+X X^3+X X^2 0 X^3+X X X^2 X^3+X^2+X X X^2 X^3+X^2 X^3 X^3+X^2+X X^2+X X^3+X X^2 X^2+X 0 X^2 X^3 X X^3+X X^3 X^2+X 0 X^3+X^2 X^3+X^2+X X^3+X X X^3+X^2 0 0 X^2+X X^3+X^2+X X^3 X^3+X^2 X X^2+X X^3+X^2 X 0 X^3+X^2 X^3+X 0 X^3 X X X^3+X X^3+X X^3+X^2+X X^3+X^2+X X^3+X^2+X X^2+X X^3+X^2+X X^3 X^3+X^2 X^3+X^2+X 0 X^3+X^2 X^3 X^3+X X X^3+X^2+X X^3+X^2 X^3+X 0 X^3+X X X X X 0 X^3+X 0 0 0 X^3 0 0 X^3 0 X^3 0 X^3 X^3 X^3 X^3 0 X^3 0 X^3 0 X^3 0 0 0 0 X^3 X^3 X^3 X^3 X^3 0 0 X^3 0 0 0 0 X^3 0 0 0 X^3 X^3 X^3 X^3 X^3 X^3 0 X^3 0 X^3 0 0 0 X^3 X^3 0 0 X^3 X^3 X^3 0 X^3 X^3 X^3 0 0 X^3 X^3 0 0 0 0 0 0 0 X^3 0 0 X^3 X^3 X^3 0 X^3 0 X^3 0 0 0 0 0 X^3 X^3 X^3 X^3 X^3 X^3 0 0 0 X^3 0 X^3 X^3 X^3 0 0 X^3 X^3 0 0 X^3 0 0 X^3 0 0 X^3 X^3 X^3 X^3 X^3 X^3 0 0 0 0 0 X^3 0 0 X^3 0 0 X^3 X^3 0 0 0 0 0 0 X^3 X^3 X^3 0 X^3 0 0 X^3 X^3 0 X^3 X^3 0 X^3 X^3 0 X^3 0 X^3 X^3 X^3 X^3 X^3 X^3 0 X^3 0 0 0 0 0 generates a code of length 86 over Z2[X]/(X^4) who´s minimum homogenous weight is 81. Homogenous weight enumerator: w(x)=1x^0+308x^81+271x^82+448x^83+302x^84+648x^85+450x^86+512x^87+267x^88+336x^89+180x^90+164x^91+34x^92+72x^93+22x^94+56x^95+3x^96+12x^97+5x^98+4x^99+1x^136 The gray image is a linear code over GF(2) with n=688, k=12 and d=324. This code was found by Heurico 1.16 in 103 seconds.