Polynomial and Cyclic Codes

Abstract

The “natural” error indicator for polynomial codes is the syndrome polynomial. The circuit presented to compute this polynomial is similar to the circuit utilized to find the redundancy of RS and BCH codes. An important family of the polynomial codes known as Fire codes is introduced. Fire codes are cyclic codes designed to correct error bursts. Thus, earlier in the chapter cyclic codes are analyzed. The generator polynomial of Fire codes is the product of two polynomials. For coding, the generator polynomial is used but decoding utilizes the two syndrome polynomials corresponding to the two factors. This approach, although apparently strange, is very compelling since all the conditions the parameter of the code must satisfy appear in a way that make clear the idea behind the code. The method however calls for some justification, provided in this chapter by the Chinese Remainder Theorem.