= Binary cyclic codes of length five
Over $\mathbb F_2$, $X^5-1=(X+1)(X^4+X^3+X^2+X+1)$ and the degree-four factor is irreducible: it has no linear root and is not divisible by the sole irreducible quadratic $X^2+X+1$. The four monic divisors give precisely four <cyclic codes>: the full five-dimensional space, the four-dimensional even-weight code, the one-dimensional repetition code, and the zero code. Their <generator polynomials of a cyclic code> are respectively $1$, $X+1$, $X^4+X^3+X^2+X+1$, and $X^5-1$.
Back to article page
