Cyclic codes

Abstract

The chapter focuses on cyclic codes. Cyclic codes are the most studied of all codes, as they are easy to encode and include the important family of Bose–Chaudhuri–Hocquenghem (BCH) codes. They are building blocks for many other codes, such as the Kerdock, Preparata, and Justesen codes. This chapter begins by defining a cyclic code to be an ideal in the ring of polynomials modulo xn – 1. A cyclic code of length n over GF(q) consists of all multiples of a generator polynomial g(x), which is the monic polynomial of least degree in the code, and is a divisor of xn – 1. The polynomial h(x) = (xn - 1)/g(x) is called the check polynomial of the code. The chapter examines that Hamming and double-error-correcting BCH codes are cyclic. The general definition of t-error correcting BCH codes over GF(q). The chapter describes techniques for encoding cyclic codes. © 1977, North-Holland Publishing Company