In coding theory, a polynomial code is a type of linear code whose set of valid code words consists of those polynomials (usually of some fixed length) that are divisible by a given fixed polynomial (of shorter length, called the generator polynomial).

Fix a finite field ${\displaystyle GF(q)}$, whose elements we call symbols. For the purposes of constructing polynomial codes, we identify a string of ${\displaystyle n}$ symbols ${\displaystyle a_{n-1}\ldots a_{0}}$ with the polynomial

Fix integers ${\displaystyle m\leq n}$ and let ${\displaystyle g(x)}$ be some fixed polynomial of degree ${\displaystyle m}$, called the generator polynomial. The polynomial code generated by ${\displaystyle g(x)}$ is the code whose code words are precisely the polynomials of degree less than ${\displaystyle n}$ that are divisible (without remainder) by ${\displaystyle g(x)}$.

Consider the polynomial code over ${\displaystyle GF(2)=\{0,1\}}$ with ${\displaystyle n=5}$, ${\displaystyle m=2}$, and generator polynomial ${\displaystyle g(x)=x^{2}+x+1}$. This code consists of the following code words:

Or written explicitly:

Since the polynomial code is defined over the Binary Galois Field ${\displaystyle GF(2)=\{0,1\}}$, polynomial elements are represented as a modulo-2 sum and the final polynomials are:

Equivalently, expressed as strings of binary digits, the codewords are:

This, as every polynomial code, is indeed a linear code, i.e., linear combinations of code words are again code words. In a case like this where the field is GF(2), linear combinations are found by taking the XOR of the codewords expressed in binary form (e.g. 00111 XOR 10010 = 10101).