The generator matrix

 1  0  0  1  1  1  0 X^3+X^2 X^3+X^2 X^3+X^2  1  1  1  1 X^2+X  1 X^2+X  1  1  1  X  1  1  X X^3+X^2+X X^2+X  1  1 X^3+X  1  1  1 X^3+X X^3+X^2+X  1  1  1  1  1 X^3+X^2  1  1 X^3 X^2+X  1 X^3  0 X^3+X^2+X  0  1 X^3+X^2+X X^3+X  1 X^3+X^2 X^3+X^2 X^2+X  1  1  0  1  1  X  1  1  1  1 X^3+X  1  1  1 X^2+X  1  1 X^3  1  1 X^3+X  1 X^3+X  1
 0  1  0  0 X^2+1 X^3+X^2+1  1  X  1  1 X^3+X^2 X^2 X^3+X^2+1 X^3+X^2+1 X^3+X^2 X+1  1 X^2+X X^3+X^2+X+1 X^3+X  1  X X^2+X  1  1  X X+1 X+1  1  1 X^3+X^2+1 X^3+X  1 X^3+X X^2+X X^3+X+1  0 X^3+X^2+1 X^2+X+1  1 X^2 X^2 X^2  1  X  1  1 X^3+X^2  1 X^3  1 X^3 X^3+X^2  1 X^3  1  1  1 X^3+X X+1 X^2+X+1  1 X^3+X^2  X X^3 X^3+X^2 X^3+X X^3+X^2+1 X^3+X+1 X^2+X+1  1 X^2+X X^2  1 X^3+X^2+X X+1  1 X^3+X  1  0
 0  0  1 X+1 X^2+X+1 X^2 X^2+X+1  1  X X^3+1  X X^3+X^2+1  1 X^2+X  1 X^2+X X^2+1  1 X+1 X^3+X X^3+X X^2 X^3+X^2+X+1 X^2 X^3+X+1  1  0  1 X+1 X^2+X+1  0  X X^3  1 X+1 X^3+X+1 X^3+X^2+X X^2+X X^2+1 X^3 X^2+1 X^3  1 X^3+X^2+X+1  1 X^3+1 X^3+X^2+X+1  1 X^2+X  X X^3+X  1 X^3+X+1 X^2+X  1 X^3+1 X^2+X X^3+X^2  1 X^2+X+1 X^2+X+1 X^3+1 X^3+X^2+X+1 X^3+X^2+X+1 X^2+X+1 X^3+X^2  1 X+1 X^2 X^3+X^2+1 X^2  0 X^3+X X^2+X+1 X^3+X^2  X X^2 X+1  X  0
 0  0  0 X^2 X^2  0 X^2 X^3+X^2 X^2 X^3 X^2  0 X^3 X^3+X^2 X^3+X^2  0 X^2 X^2  0  0 X^3 X^3+X^2  0 X^2 X^3 X^3 X^3 X^3+X^2 X^3+X^2  0 X^3+X^2 X^2 X^3 X^3+X^2 X^3+X^2 X^2  0  0  0 X^3+X^2 X^3+X^2 X^2 X^3  0  0 X^3+X^2 X^3 X^3  0 X^3+X^2 X^2 X^2 X^3+X^2 X^3 X^2  0 X^3 X^2 X^2 X^3 X^3+X^2 X^3+X^2 X^2 X^2 X^3 X^3+X^2 X^2 X^3+X^2 X^2 X^2 X^3+X^2 X^2 X^3 X^3+X^2  0 X^2 X^3+X^2 X^3  0  0

generates a code of length 80 over Z2[X]/(X^4) who´s minimum homogenous weight is 74.

Homogenous weight enumerator: w(x)=1x^0+127x^74+838x^75+1158x^76+1666x^77+1965x^78+1954x^79+1972x^80+1874x^81+1452x^82+1120x^83+771x^84+694x^85+306x^86+184x^87+136x^88+82x^89+45x^90+30x^91+1x^92+4x^93+1x^94+2x^95+1x^96

The gray image is a linear code over GF(2) with n=640, k=14 and d=296.
This code was found by Heurico 1.16 in 5.42 seconds.