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 X^3 1 X 1 1 1 1 1 1 1 1 1 0 X 1 X 1 X^2 1 X 1 X 1 1 1 1 X^2 1 0 1 0 X X 1 1 1 X 0 X 0 X^3+X^2+X X^2 X^2+X X^3+X^2 X 0 X^2+X X^3 X^2+X X^3+X X^2 X^2 X 0 X^2+X X^2 X^3+X X^3 X^3+X X^3 X^3+X X^2+X 0 X^3 X X^2+X X^3 X^2+X 0 X^2 X^3+X X^3+X^2+X 0 0 X X^3+X^2 X^3+X X X X^3 X^2+X X^3+X^2+X X^2+X X^3+X^2 0 X^2 X^3+X^2+X X^3+X^2+X X^2 X^2 X X^3+X^2+X X X^3+X X^2 X^3 X^2 X X^2 X^2+X X X^3+X^2+X 0 X^3+X^2 X 0 X X X X^2+X X^2+X 0 X^3 X^3+X^2 X^3+X^2+X 0 0 X^3+X^2 0 X^2 0 X^3 0 X^2 X^2 X^3 X^3+X^2 X^3+X^2 X^3+X^2 0 X^2 0 X^3+X^2 X^3 X^3+X^2 X^2 X^3 X^2 0 0 0 X^2 0 X^2 X^3 X^2 X^2 X^3 X^2 X^3 0 X^3 0 X^2 X^3 X^3 X^2 0 X^2 X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^2 X^3+X^2 0 0 X^3 X^2 X^2 X^2 X^3+X^2 X^3+X^2 X^2 X^2 X^2 X^2 0 X^3 0 X^3 X^3+X^2 X^3 X^3 0 X^2 X^3+X^2 0 X^2 X^3 X^3+X^2 X^3+X^2 X^2 0 0 0 X^3+X^2 0 X^3 X^3 X^2 X^2 X^2 X^2 0 0 X^2 X^3+X^2 X^2 X^3 X^3+X^2 X^3+X^2 X^3 0 X^3+X^2 X^3+X^2 X^3 0 X^3+X^2 0 X^3+X^2 X^3 0 X^2 X^2 X^3 X^3+X^2 X^2 X^2 X^2 0 0 X^2 0 X^3 0 X^3+X^2 X^3 X^3+X^2 X^3+X^2 X^3 X^3+X^2 X^3 0 X^2 X^3+X^2 X^3+X^2 X^3 0 X^2 X^3 X^3 X^3 0 X^2 X^3 X^3 X^3 X^3 X^3+X^2 X^3+X^2 0 X^3 0 X^3+X^2 X^3+X^2 X^2 0 X^2 0 X^3 0 0 0 0 X^3 X^3 X^3 X^3 0 0 0 X^3 0 X^3 X^3 X^3 0 0 X^3 0 0 X^3 0 0 X^3 0 X^3 0 X^3 X^3 X^3 X^3 0 0 X^3 0 X^3 X^3 0 0 0 0 X^3 X^3 0 X^3 0 X^3 X^3 X^3 0 X^3 0 0 X^3 X^3 X^3 X^3 X^3 0 X^3 0 X^3 X^3 0 0 X^3 X^3 0 X^3 0 X^3 0 0 X^3 X^3 X^3 0 generates a code of length 78 over Z2[X]/(X^4) who´s minimum homogenous weight is 72. Homogenous weight enumerator: w(x)=1x^0+147x^72+144x^73+229x^74+292x^75+496x^76+440x^77+640x^78+536x^79+460x^80+224x^81+161x^82+132x^83+118x^84+24x^85+16x^86+18x^88+8x^90+4x^92+2x^94+3x^96+1x^128 The gray image is a linear code over GF(2) with n=624, k=12 and d=288. This code was found by Heurico 1.16 in 1.23 seconds.