The generator matrix

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

generates a code of length 78 over Z2[X]/(X^4) who´s minimum homogenous weight is 73.

Homogenous weight enumerator: w(x)=1x^0+126x^73+609x^74+1040x^75+1324x^76+986x^77+1028x^78+700x^79+741x^80+458x^81+429x^82+332x^83+199x^84+114x^85+56x^86+16x^87+21x^88+6x^90+4x^93+1x^96+1x^100

The gray image is a linear code over GF(2) with n=624, k=13 and d=292.
This code was found by Heurico 1.16 in 2.61 seconds.