Reed-Solomon Codes

1. Introduction

Reed-Solomon codes are block-based error correcting codes with a wide range of applications in digital communications and storage. Reed-Solomon codes are used to correct errors in many systems including:

• Storage devices (including tape, Compact Disk, DVD, barcodes, etc)
• Wireless or mobile communications (including cellular telephones, microwave links, etc)
• Satellite communications
• Digital television / DVB
• High-speed modems such as ADSL, xDSL, etc.

A typical system is shown here:

The Reed-Solomon encoder takes a block of digital data and adds extra "redundant" bits. Errors occur during transmission or storage for a number of reasons (for example noise or interference, scratches on a CD, etc). The Reed-Solomon decoder processes each block and attempts to correct errors and recover the original data. The number and type of errors that can be corrected depends on the characteristics of the Reed-Solomon code.

2. Properties of Reed-Solomon codes

Reed Solomon codes are a subset of BCH codes and are linear block codes. A Reed-Solomon code is specified as RS(n,k) with s-bit symbols.

This means that the encoder takes k data symbols of s bits each and adds parity symbols to make an n symbol codeword. There are n-k parity symbols of s bits each. A Reed-Solomon decoder can correct up to t symbols that contain errors in a codeword, where 2t = n-k.

The following diagram shows a typical Reed-Solomon codeword (this is known as a Systematic code because the data is left unchanged and the parity symbols are appended):

Example:

A popular Reed-Solomon code is RS(255,223) with 8-bit symbols. Each codeword contains 255 code word bytes, of which 223 bytes are data and 32 bytes are parity. For this code:

n = 255, k = 223, s = 8

2t = 32, t = 16

The decoder can correct any 16 symbol errors in the code word: i.e. errors in up to 16 bytes anywhere in the codeword can be automatically corrected.

Given a symbol size s, the maximum codeword length (n) for a Reed-Solomon code is n = 2^s – 1

For example, the maximum length of a code with 8-bit symbols (s=8) is 255 bytes.

Reed-Solomon codes may be shortened by (conceptually) making a number of data symbols zero at the encoder, not transmitting them, and then re-inserting them at the decoder.

Example

: The (255,223) code described above can be shortened to (200,168). The encoder takes a block of 168 data bytes, (conceptually) adds 55 zero bytes, creates a (255,223) codeword and transmits only the 168 data bytes and 32 parity bytes.

The amount of processing "power" required to encode and decode Reed-Solomon codes is related to the number of parity symbols per codeword. A large value of t means that a large number of errors can be corrected but requires more computational power than a small value of t.

Symbol Errors

 

One symbol error occurs when 1 bit in a symbol is wrong or when all the bits in a symbol are wrong.

Example

: RS(255,223) can correct 16 symbol errors. In the worst case, 16 bit errors may occur, each in a separate symbol (byte) so that the decoder corrects 16 bit errors. In the best case, 16 complete byte errors occur so that the decoder corrects 16 x 8 bit errors.

Reed-Solomon codes are particularly well suited to correcting burst errors (where a series of bits in the codeword are received in error).

Decoding

 

Reed-Solomon algebraic decoding procedures can correct errors and erasures. An erasure occurs when the position of an erred symbol is known. A decoder can correct up to t errors or up to 2t erasures. Erasure information can often be supplied by the demodulator in a digital communication system, i.e. the demodulator "flags" received symbols that are likely to contain errors.

When a codeword is decoded, there are three possible outcomes:

1. If 2s + r < 2t (s errors, r erasures) then the original transmitted code word will always be recovered,

OTHERWISE

2. The decoder will detect that it cannot recover the original code word and indicate this fact.

OR

3. The decoder will mis-decode and recover an incorrect code word without any indication.

The probability of each of the three possibilities depends on the particular Reed-Solomon code and on the number and distribution of errors.

Coding Gain

 

The advantage of using Reed-Solomon codes is that the probability of an error remaining in the decoded data is (usually) much lower than the probability of an error if Reed-Solomon is not used. This is often described as coding gain.

Example:

A digital communication system is designed to operate at a Bit Error Ratio (BER) of 10^-9, i.e. no more than 1 in 10⁹ bits are received in error. This can be achieved by boosting the power of the transmitter or by adding Reed-Solomon (or another type of Forward Error Correction). Reed-Solomon allows the system to achieve this target BER with a lower transmitter output power. The power "saving" given by Reed-Solomon (in decibels) is the coding gain.