Reciprocal polynomial ($p^{\ast }(X)=X^{\mathrm{deg}p}p(X^{-1})$)

The reciprocal polynomial reverses the coefficients of a polynomial with nonzero constant term. For a binary cyclic code with generator $g$ and canonical check polynomial $h$, the reciprocal $h^{\ast }$ generates the dual of a cyclic code.

// Target: algebra.bigb

Let $P$ be a nonzero polynomial of even degree $2m$ over a field of characteristic different from two, satisfying the displayed reciprocal identity. If $P(x)=(x-1{)}^{r}A(x)$ with $A(1)\ne 0$, cancellation gives $A(x)=(-1{)}^r{x}^{2m-r}A(1/x)$. Substitution of $x=1$ forces $r$ even. If $P(x)=(x+1{)}^{s}B(x)$ with $B(-1)\ne 0$, cancellation gives $B(x)=x^{2m-s}B(1/x)$; substitution of $x=-1$ forces $s$ even. Zero multiplicities are included. Pairing nonfixed reciprocal roots alone cannot prove this parity, because $1$ and $-1$ are their own inverses.