 Solution

Over ${\mathbb{F}}_2$, subtraction equals addition. The polynomial $X^7-1$ has the factorizationThe two cubic factors have no root in ${\mathbb{F}}_2$, so they are irreducible, and they are reciprocal to one another. PutEvery cyclic code corresponds to one monic divisor of $X^7-1$. The complete list isThus the two cubic generators give the two cyclic coordinate orientations of Hamming's $[7,4,3]$ code. Their duals have generators $(X+1)f$ and $(X+1)f^{\ast }$ and are the $[7,3,4]$ simplex codes. The remaining four are the familiar whole-space, even-weight, repetition, and zero codes. This is the complete classification of binary cyclic codes of length seven.