Binarylinearcodes
2.1Theconceptofbinarylinearcodes
Exercises2.1
2.1.1. Showthatourcode [7, 4, 3]2 isperfect.
Wehave V2(1, 7)=8 Thespherepackingboundsays M 8 ≤ 27 , whichis satisfiedwithequality.
2.1.2. Trytodecideifan [8, 4, 4]2 exists.
Startfromageneratormatrixofthe[7, 4, 3]2. Appendanewbittoeach rowsuchthattheweightofeachrowiseven.Theresultisageneratormatrix ofan[8, 4, 4]2
2.1.3. Giveanexampleshowingthatthebasisofacode isnotuniquelydetermined.
Anyoldexamplewilldo.
2.1.4. Determinetheparametersofthebinarylinearcodegeneratedbythe rowsofthematrix
1001101 0101011 0010111
Simplywriteoutallcodewords: 0000000 1001101 0101011 0010111 1100110 1011010 0111100 1110001
18 IntroductiontoCodingTheory,SecondEditionSolutionsManual
Theseare8=23 differentcodewords.Eachnonzerocodewordhasweight 4. Inparticular d =4. Theparametersare[7, 3, 4]2.
2.1.5. Computetheparameters [n,k,d]2 ofthebinarylinearcodegenerated by G =
1100001101
0011001011
0000110111
Findanonzerocodewordofminimumweight.
Againwriteoutallcodewords:
0000000000
1100001101
0011001011
0000110111
1111000110
1100111010
0011111100
1111110001
Againwehave8=23 codewords.Thesmallestnonzeroweightis d =5 Theparametersare[10, 3, 5]2 Thesecondwordinthelisthasweight5
2.2Blockcoding
Exercises2.2
2.2.1. Followthroughallstagesofblockcodingfortheinputstring 00101110 whentransmissionerrorsoccurincoordinates 3, 7 and 14. Nothingtosayhere.Justdoit.
2.3Theeffectofcoding
Exercises2.3
2.3.1. Computetheblockerrorprobabilityoftherepetitioncodeoflength 3, seenasacodeencodingblocksoflength 4 intomessagesoflength 12. Forexample,0010isencoded000000111000 Nodecodingerrormeansno errorineachofthefourblocksoflength3 Foreachblocktheprobabilityof noerroris(1 p)3 +3p(1 p)2 ≈ 1 3p2 Thetotalprobabilityofnodecoding erroristherefore
andtheerrorprobabilityis P ≈ 18p2 Thisisveryclosetothecaseofthe binaryHammingcode,whichhasamuchhigherinformationrate.
2.3.2. Computeinformationrateandblockerrorprobabilityforacode [9, 5, 3]2 (itexists).
Informationrate R =5/9, probabilityofcorrectdecoding
Blockerrorprobability P ≈ 36p2
2.3.3. Computeinformationrateandblockerrorprobabilityforacode [23, 12, 7]2 (the binaryGolaycode).
R =12/23. Probability1 P ofnoerror (1 p)23 +23p(1 p)22 + 23 2 p 2(1 p)21 + 23 3 p 3(1 p)20 whichis ≈ 1 23 4 p4 . Theerrorprobabilityis ≈ 8855p4 .
2.3.4. Showthattheblockerrorprobabilityofa [n,k, 2e +1]2-code isbounded(approximately)by n e +1 pe+1
Thisgeneralizesourearliercalculations.Theprobability1 P ofcorrect decodingisatleast e i=0 n i pi(1 p)n i (thereare n i errorpatternswith precisely i errors,theprobabilityofeachsuchpatternis pi(1 p)n i).Seethis expressionasapolynomialin p. Theleadingterm(theconstantterm)isof course1. Fixsomeexponent j. Thecoefficientof pj is e i=0 n i ( 1)j i n i j i .
20 IntroductiontoCodingTheory,SecondEditionSolutionsManual
Ourinterpretationofbinomialnumbersintermsofsubsetsshowsthefollowing identity:
n i n i j i = n j j i (thesearetwowaysofcountingpairsofdisjointsubsets,oneofcardinality i, theotherofcardinality j i, ofan n-set).Ourcoefficientistherefore n j e i=0 ( 1)j i j j i
For j =0theresultis1confirmingthattheleadingtermis1. For1 ≤ j ≤ e thesumsimplyisthebinomialexpansionof(1 1)j =0, showingthatthose powers pj donotoccur.For j = e +1wehave e i=0( 1)e+1 i e+1 e+1 i = (1 1)e+1 1= 1.
2.4Duality
Exercises2.4
2.4.1. Whenistheall-1-wordorthogonaltoitself?
Theall 1-wordisorthogonaltoitselfifthelength n iseven.
2.4.2. Acodeis self-dual ifitequalsitsdual.Isthereaself-dual [6, 3, 3]2?
Thereisnoself-dual[6, 3, 3]2
Asinparticulareachcodewordmustbeorthogonaltoitself,allweightsmust beeven.Wewouldhaveacode[6, 3, 4]2 Theall 1-wordisorthogonaltothe codeandthereforecontainedinthecode.Allothercodewordshaveweight 4. However,thesumofawordofweight4andtheall 1-wordhasweight2, contradiction.
2.4.3. Finda (4, 8)-matrixinstandardform(startingwiththeunitmatrix I) whichgeneratesaself-dualcode(C⊥ = C)withparameters [8, 4, 4]2.
10000111 01001011 00101101 00011110
2.4.4. FindageneratormatrixoftheHammingcode [7, 4, 3]2 instandard form.UsetheP-transformtofindacheckmatrix.
Binarylinearcodes 21
Inordertofindageneratormatrixinstandardformsimplypickthose codewordsof H startingwiththequadruplesofweight1. Thisyieldsthe generatormatrix
1000011 0100101 0010111 0001110
TheP-transformyieldsthecheckmatrix
0111100 1011010 1110001
2.4.5. Isthereaself-dual [12, 6, 6]2-code?
Writeageneratormatrixinstandardform G =(I|P )where P isa(6, 6) matrix. Becauseofself-dualitytherowsof P haveoddweight,becauseof d = 6theyhaveweight5. Thesumoftworowsof G hasweight ≤ 4, contradiction.
2.4.6. Istherea [12, 6, 6]2-code?
No.Afterpuncturing(cancelthelastcoordinateineachcodeword)an [11, 6, 5]2-codeisobtained.Itcontradictsthesphere-packingbound.
2.5BinaryHammingandSimplexcodes
Exercises2.5
2.5.1. Usingmatrix M3, findatleast 5 differentcodewordsintheHamming code [7, 4, 3]2.
ThecodewordsintheHammingcodegeneratedbymatrix M3 :therows 1001101, 0101011, 0010111, thepairwisesums1100110, 1011010, 0111100 and0000000, 1110001
2.5.2. Use M4 andfindatleast 5 differentcodewordsin H4(2).
Similarlyusing M4 :thereare16codewords.
2.5.3. UsethebinaryHammingcode [7, 4, 3]2. Decodethereceivedvectors
y1 =(1, 1, 0, 1, 1, 0, 0),y2 =(1, 1, 1, 1, 1, 1, 1), y3 =(1, 1, 1, 0, 0, 0, 0).
IntroductiontoCodingTheory,SecondEditionSolutionsManual
Decoding:
2.5.4. Provebyinductionon r thateachnontriviallinearcombinationofthe rowsof Mr (eachnonzerowordoftheSimplexcode Sr(2))hasweight 2r 1 .
For r =1and r =2thisisimmediatelychecked.Let r> 2 Orderthe columnsof Mr (thisorderingisimmaterialforourproblem)suchthatinthe firstrowwehaveallzeroesontheleft,theonesontheright.Then Mr has theform Mr = 0 1 1 Mr 1 0 Mr 1 (inthemiddlethereisacolumn(1, 0)t).Eachlinearcombinationnot involvingthefirstrowhasweight2r 2 +2r 2 =2r 1 . Addingthefirstrow yieldsweight2r 2 +1+(2r 1 1 2r 2)=2r 1 . Fortunatelyalsothefirst rowhasweight2r 1
2.5.5. ShowthatthebinarySimplexcodes Sr(2),r ≥ 3 areself-orthogonal (containedintheirorthogonals).
Considerthegeneratormatrix Mr Eachcodewordisorthogonaltoitself asithasevenweight2r 1 Comparerow i androw j of Mr Thenumberof coordinateswherebothhaveentry1is2r 2 , whichiseven.
2.6Theprincipleofdualityforbinarylinearcodes
Exercises2.6
2.6.1. Define q-aryorthogonalarraysforarbitrary q.
Anarraywith n columnsandentriesfroma q-setisa q-aryorthogonal array ofstrength t ifintheprojectionontoanysetof t columnseach t-tuple ofentriesoccursthesamenumber λ oftimes.Wewritetheparametersas
OAλ(t,n,q)
2.6.2. Showthatabinaryorthogonalarrayofstrength t> 1 alsohasstrength t 1.
Let A beabinaryorthogonalarrayofstrength t> 1. Consideranysetof t 1columnsandembeditinasetof t columns.Becauseofstrength t there isanumber λ suchthatevery t-tupleoccurs λ timesintheprojectiononto the t columns.Thisshowsthatevery(t 1)-tupleoccurs2λ timesinthe projectionontothe t 1columns.
2.6.3. Findacheckmatrixof S3(2) byapplyingtheP-transformto M3 Use thischeckmatrixtoprovethat S3(2) hasminimumweight 4
ApplicationoftheP-transformto M3 yieldsasgeneratormatrixofthe Hammingcode,checkmatrixoftheSimplexcode S3(2), thematrix
110 1000 101 0100 011 0010 111 0001
Inordertoprovethat S3(2)hasminimumdistance4wehavetoseethatno 3orlesscolumnsof H addupto0 Thereisno0-column,andthereareno twoequalcolumns.Assume3columnsaddto0 Clearlytheyarenotallin therightsection,correspondingtotheunitmatrix.Iftwoareontheright, theonecolumnontheleftmusthaveweight2, contradiction.Ifoneison theright,twoontheleftmustaddtoaweight1column.Thisisnotthe case.Theremainingcaseisthatallareontheleft,butthesumofthe3left columnsis(0, 0, 0, 1)t , contradiction.
2.6.4. Showthateachperfectbinarylinearcodeofdistance d =3 hasthe parametersofoneofthebinaryHammingcodes.
Wehave2n =2k(1+ n) Thisshowsthatthelength n hastheform n = 2r 1 Thisshows r + k = n,k =2r 1 r.
2.6.5. Showthateachperfectbinarylinearcodeofdistance d =3 isequivalent tooneofthebinaryHammingcodes.
Continuingfromthepreviousexerciseweseethatacheckmatrixisan (r, 2r 1)-matrix H whichhasnozerocolumn(because d> 1)andnorepeated columns(as d> 2).Itfollowsthatthecolumnsof H arepreciselyallnonzero r-tuplesinsomeorder.
2.6.6. Describean OA1(n 1,n, 2) forarbitrarylength n.
Thisisthelinearsumzerocode:useasrowsallbinary n-tuplesofeven weight.