Cyclic codes are codes that are preserved under a cyclic permutations of coordinates.
More precisely, a cyclic code is a set \(C \subset \mathbb{F}_q^n\) such that \((c_1, c_2 \dots c_{n-1}, c_n ) \in C \Rightarrow (c_n , c_1 , c_2 \dots c_{n-1}) \in C\). The cyclic code is linear if it is also a vector subspace (or just closed under addition, really) when \(\mathbb{F}_q^n\) is seen as an \(\mathbb{F}_q\)-vector space. If we model \(\mathbb{F}_q^n \simeq \frac{\mathbb{F}_q[x]}{\langle x^n - 1\rangle}\) (by just mapping vectors on the coefficients), then we have this very useful lemma.
A subset \(C \subset \frac{\mathbb{F}_q[x]}{\langle x^n -1 \rangle}\) is a linear cyclic code if and only if it is an ideal inside \(\frac{\mathbb{F}_q[x]}{ \langle x^n -1\rangle}\).
Being \(C\) being linear is equivalent to \(C\) being an \(\mathbb{F}_q\)-subspace. Being cyclic gives us that \(f \in C \Rightarrow xf \in C\). This means that for any polynomial \(g= \sum_{i=0}^{n-1} g_i x^{i}\), \(g f \in C\). Hence it is an ideal.
On the other hand, being an ideal implies being linear and closed under multiplication with \(x\).
The code in Example can be presented as the ideal \(\langle 1 + x\rangle \subset \frac{\mathbb{F}_2[x]}{\langle x^{3} - 1 \rangle}\).
Is there a sequence of asymptotically good cyclic codes?
See [1] for a good introduction and some curious connections.
Counting this is the same as counting the number of ideals in \(\mathbb{F}_q[x]/\langle x^{n} -1\rangle\). This is also then the same problem as counting the number of irreducible factors of \(x^{n}-1\) in \(\mathbb{F}_q[x]\).
This page was updated on November 13, 2023.
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