Chapter 3: Cyclic Codes over Finite Fields

A linesr code C is called a cyclic code if c = (c₀, c₁, …, c_n−1) ∊ C implies (c_n−1, c₀, c₁, …, c_n−2) ∊ C. As a subclass of linear codes, cyclic codes have wide applications in consumer electronics, data storage systems, and communication systems as they have some efficient encoding and decoding algorithms. In this chapter, we introduce the basic theory of cyclic codes over finite fields without providing a proof in many cases. We refer the reader to Huffman and Pless (2003)[Chapter 4] for a proof of such result.