Repeated-root cyclic code

A repeated-root cyclic code has length divisible by the characteristic of its field, so the polynomial $X^N-1$ has repeated factors. Its generator can involve powers of those factors. In characteristic two, length $2^t$ gives $X^{2^t}+1=(X+1{)}^{2^t}$.

// Target: algebra.bigb

If the generator polynomial of a cyclic code has distinct roots ${\alpha }_i$ with multiplicities $m_i$, a word belongs to the code exactly when $c^{[\ell ]}({\alpha }_i)=0$ for $0\le \ell <m_i$, where $c^{[\ell ]}$ is its Hasse derivative. The corresponding rows have entries $\left(\genfrac{}{}{0ex}{}{j}{\ell }\right){\alpha }_i^{j-\ell }$ for $j\ge \ell$ and zero otherwise. Expansion in a basis of the splitting field gives equations over the original finite field; an independent selection has rank ${\sum }_i{m}_i$.

For $N=2^t$, the binary cyclic codes are exactly the nested ideals generated by $(X+1{)}^j$ for $0\le j\le N$. They have dimensions $N-j$ and canonical check polynomials $(X+1{)}^{N-j}$. The endpoint ideals are the entire word space and the zero code.

// Target: algebra.bigb