BCH code

Overview

BCH codes are a class of error correcting codes that were invented in 1959 by French mathematicians Alexis Hocquenghem and Raj Bose, along with Dwijendra Kumar Ray-Chaudhuri. BCH codes have the ability to correct multiple errors and are widely used in applications such as satellite communication, compact disc players, and data transmission systems.

History

In 1959, Alexis Hocquenghem, a French mathematician, introduced BCH codes. Later in the same year, Raj Bose and D.K. Ray-Chaudhuri, two Indian mathematicians, independently discovered the same codes. As a result, these codes are known as BCH codes, an acronym derived from the initials of their last names.

Definition

A BCH code is a type of cyclic error-correcting code constructed using polynomials over a finite field (also known as a Galois field). The BCH code is defined by its length (n), the number of error bits it can correct (t), and the size of the field (m). The length of the code is usually represented as n = 2^m - 1.

A photograph of a binary BCH code example written on a whiteboard.

Construction

The construction of BCH codes involves the generation of a generator polynomial, which is used to encode the data. This polynomial is a factor of the polynomial x^n - 1, where n is the length of the code. The roots of the generator polynomial are a set of consecutive powers of a primitive element in the Galois field.

Decoding

The decoding of BCH codes involves two main steps: error location and error correction. The error location is determined using the error locator polynomial, which is derived from the syndrome calculations. Once the error locations are known, the errors can be corrected.

Applications

BCH codes are widely used in digital communication and data storage systems. They are particularly effective in applications where bursts of errors are likely to occur. Some of the common applications of BCH codes include satellite and wireless communication, compact disc players, and flash memory storage devices.

Advantages and Limitations

One of the main advantages of BCH codes is their ability to correct multiple errors. They are also relatively easy to implement in hardware and software. However, BCH codes have their limitations. The decoding process can be complex and computationally intensive, particularly for long code lengths and high error correction capability. Furthermore, BCH codes are not always optimal in terms of code rate and minimum distance.