Codeword¶
Given a vector \(a = (a_1,\dots,a_n) \in \mathbb{F}_p^n\), \(q(x)\) is the low-degree extension of \(a\). Consider the vector \(LDE(a)\) of length \(p = |\mathbb{F}_p|\) whose \(i\)-th entry is \(q(i)\). If \(p\gg n\), then \(LDE(a)\) is vastly longer than \(a\) itself and contains \(a\) as a sub-vector. In the coding theory literature, \(a\) is referred to as a message, and the encoding \(LDE(a)\) is called the codeword corresponding to message \(a\).