Reed Solomon

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PROJECT ON :

REED SOLOMON

CODE

By:

Anuj Gupta Gopal Krishan

100201 100209

Under the guidance of

Mr. Puli Kishore Kumar

NIT Delhi


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CONCURRENT ERROR DETECTION

IMPLEMENTATION


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DEFINATION:

Reed

Solomon RS) codes are non-

binary cyclic error-correcting codes invented

by Irving S. Reed and Gustave Solomon. Theydescribed a systematic way of building codes

that could detect and correct

multiple random symbol errors. By

adding t check symbols to the data, an RScode can detect any combination of up

to terroneous symbols, or correct up to t/2

symbols.


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The ReedSolomon code is a [n, k, nk+ 1] code;

in other words, it is a linear block code of

length nwith dimension kand minimum Hamming

distance n k + 1. The ReedSolomon code is

optimal in the sense that the minimum distance

has the maximum value possible for a linear code

of size (n, k); this is known as the Singleton bound.Such a code is also called a maximum distance

separable (MDS) code.


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In digital communication, Reed-Solomon (RS)

codes refer to as a part of channel coding thathad becoming very significant to better

withstand the effects of various channel

impairments such as noise, interference and

fading. This signal processing technique isdesigned to improve communication

performance and can be deliberate as

medium for accomplishing desirable system

trade-offs.


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Galois field arithmetic is used for encoding and

decoding of Reed Solomon codes. Galois fieldmultipliers are used for encoding the information

block. The encoder attaches parity symbols to the

data using a pre-determined algorithm before

transmission. At the decoder, the syndrome of thereceived codeword is calculated. VHDL impleme -

ntation creates a flexible, fast method and high

degree of parallelism for implementing the Reed

Solomon codes. Computer simulation tool and

MATLAB will be used to create and run extensively

the entire simulation model for performance

evaluation and VHDL is used to implemented the

design of RS encoder.


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Objective :

To analyze the important characteristics of RS coding

techniques that could be used for error control in a

communication system for reliable transmission of digital

information over the channel.

To study the Galois Field Arithmetic on which the most

important and powerful ideas of coding theory are based.

To study the Reed Solomon codes and the various

methods used for encoding and decoding of the codes to

achieve efficient detection and correction of the errors.Implementation of the Reed Solomon codes in MatLab

& RS encoder in VHDL.


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Application :

Reed Solomon codes are error correcting codes thathave found wide ranging applications throughout the

fields of digital communication and storage. Some of

which include :

Storage Devices (hard disks, compact disks, DVD,

barcodes)

Wireless Communication (mobile phones,

microwave links).

Digital Television

Broadband Modems (ADSL, X DSL, etc).Deep Space and Satellite Communications

Networks (CCSDS).


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07 FEB,2013


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These are the simplest form of error detecting codes,

with a hamming distance of two (d=2),and a single

check bit (irrespective of the size of input data). They

are of two basic types: Odd and Even. For an even-

parity code the check bit is defined so that the total

number of 1s in the code word is always even; for an

odd code, this total is odd. So, whenever a fault

affects a single bit, the total count gets altered and

hence the fault gets easily detected. A majordrawback of these codes is that their multiple fault

detection capabilities are very limited.

PARITY CHECK


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In these codes the summation of all the information

bytes is appended to the information as b-bit

checksum. Any error in the transmission will be

indicated as a resulting error in the checksum. This

leads to detection of the error. When b=1, these codes

are reduced to parity check codes. The codes are

systematic in nature and require simple hardware

units.

CHECKSUM CODES


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In this scheme the codeword is of a standard weight m

and standard length n bits. Whenever an error occurs

during transmission, the weight of the code word

changes and the error gets detected. If the error is a 0

to 1 transition an increase in weight is detected,

similarly 1 to 0 leads to a reduction in weight of the

code, leading to easy detection of error. This scheme

can be used for detection of unidirectional errors,

which are the most common form of error in digitalsystems.

M-OUT-OF-N CODES


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Berger codes are systematic unidirectional error

detecting codes. They can be considered as an

extension of the parity codes. Parity codes have one

check bit, which can be considered as the number of

information bits of value 1 considered in modulo 2. On

the other hand Berger codes have enough check bits

to represent the count of the information bits having

value 0. The number of check bits (r) required for k -bit

information is given by

r=[log2(k-1)]

BERGER CODES


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ERROR CORRECTING CODE


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BCH codes are the most important and powerful classes of l inearblock codes which are cyclic codes with a wide variety ofparameters. The most common BCH codes are characterized asfollows. Specifically, for any positive integer m (equal to or greaterthan 3) and t [less than (2m 1) / 2 ] there exists a binary BCHcode with the following parameters:

Block length: = 2m 1 n

Number of message bits k n mt

Minimum distance 2 1 min d t +

Where m is the number of parity bits and t is number of errors thatcan be corrected. Each BCH code is a t error correcting code inthat it can detect and correct up to t random errors per codeword.

The Hamming single error correcting codes can be described asBCH codes. The BCH codes offer f lexibil ity in the choice of codeparameters, block length and code

rate.

BOSE CHAUDHURI HOCQUENQHEM

(BCH) CODES


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Hamming codes can also be defined over the non

binary field. The parity check matrix is designed by

setting its columns as the vectors of GF(p) m whose

first non zero element equals one. There are

n = ( pm 1) /( p 1) such vectors and any pair of

these is linearly independent.

HAMMING SINGLE ERROR CORRECTING

CODES


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Burst errors mean that a no. of errors occur simultaneously in

a code. If a particular symbol is in error, then the chances are

good that its immediate neighbors are also wrong. Burst

errors occur for instance in mobile communications due to

fading and in magnetic recording due to media defects. Reed Solomon code is one of the most important type of

burst-error correcting codes.

BURST ERROR CORRECTING CODES

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Reed Solomon