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CODING THEORYCODING THEORY

A Birds Eye View : ContinuedA Birds Eye View : Continued

Block Codes: BasicsBlock Codes: Basics

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Types of Channel CodesTypes of Channel Codes

lock Codeslock Codes ( Codes with( Codes with strong algebraic flavorstrong algebraic flavor))~1950~1950------ Hamming CodeHamming Code ((Single error correctionSingle error correction))All codes in 50s were too weak compared to theAll codes in 50s were too weak compared to thecodes promised by Shannoncodes promised by Shannon

Major Breakthrough..1960Major Breakthrough..1960

BCH CodesBCH Codes

ReedReed--Solomon CodesSolomon Codes

Capable of correctingCapable of correctingMultiple ErrorsMultiple Errors

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ConvolutionalConvolutional CodesCodes

Codes withCodes with Probabilistic flavor.Probabilistic flavor.

Late 1950s but gained popularity after theLate 1950s but gained popularity after theintroduction ofintroduction of Viterbi algorithmViterbi algorithm in 1967.in 1967.

Developed from the idea of sequential decodingDeveloped from the idea of sequential decoding NonNon--block codesblock codes

Codes are generated by a convolution operation onCodes are generated by a convolution operation onthe information sequencethe information sequence

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Coding Schemes: TrendCoding Schemes: Trend

Since 1970s the two avenues of research startedSince 1970s the two avenues of research started

working togetherworking together This resulted in the development towards theThis resulted in the development towards the

codes promised by Shannoncodes promised by Shannon TodayToday Turbo Codes,Turbo Codes, are capable of achievingare capable of achieving

an improvement close to Shannon Limitan improvement close to Shannon Limit

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Coding SchemesCoding Schemes

Applications demand for wide range of dataApplications demand for wide range of data

rates, block sizes, error rates.rates, block sizes, error rates. No single error protection scheme works for allNo single error protection scheme works for all

applications.applications. Some requires the use of multipleSome requires the use of multiple

coding techniques.coding techniques.

A common combination uses anA common combination uses an innerinner

convolutional codeconvolutional codeand anand an outerouterReedReed--Solomon code.Solomon code.

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16Slide 3

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Block codes: basic deBlock codes: basic defifinitionsnitions AnAn alphabetalphabet is a discrete (usuallyis a discrete (usually fifinite) set ofnite) set of

symbolssymbols..

example:example: BB == {{ 00;; 11}} is the binary alphabetis the binary alphabet

DeDefifinitionnition: A: A block codeblock code ofof blocklengthblocklength nn over anover an

alphabetalphabet XX is a nonemptyis a nonempty set ofset of nn--tuplestuples ofofsymbols fromsymbols from XX..

TheThe nn--tuplestuples of the code are calledof the code are called code wordscode words..

Code words areCode words are vectorsvectors whose components arewhose components aresymbols insymbols in XX..

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Block codes: basic deBlock codes: basic defifinitionsnitions Code words of lengthCode words of length nn are typically generatedare typically generated

by encoding messages ofby encoding messages of kk informationinformation bitsbitsusing an invertible encoding function.using an invertible encoding function.

Number ofNumber of codewordscodewords isis MM == 22kk ,, RateRate RR == k/nk/n

The rate is a dimensionless fractionThe rate is a dimensionless fraction;; the fractionthe fractionof transmitted symbols thatof transmitted symbols that carry information.carry information.

A code withA code with blocklengthblocklength nn and rateand rate kk//nn is calledis calledanan ((n; kn; k)) cocodede

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Systematic encoderSystematic encoder The error protection ability of a block codeThe error protection ability of a block code

depends only on thedepends only on the setset ofof codewordscodewords, not on, not onthe mapping from source messages tothe mapping from source messages tocodewordscodewords..

An encoder isAn encoder is systematicsystematic when it copies thewhen it copies the kkmessage symbols unchanged into themessage symbols unchanged into thecodeword.codeword. CodewordsCodewords are of the formare of the form

cc = [= [mm pp ]] oror cc = [= [ p mp m]]wherewhere mm is the vector ofis the vector of kk message symbolsmessage symbolsandand pp is the vector ofis the vector of nn--kk redundantredundant oror checkcheck

symbols.symbols.

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Linear Block CodesLinear Block Codes

matrixGeneratorG

(vector)wordmessagem

(vector)wordcode

,

=

c

where

Gmc