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136 CHAPTER 7 ALGEBRAIC CODING THEORYgenerates a Hamming code. What are the error-correcting propertiesof a Hamming code?(b) The column corresponding to the syndrome also marks the bit thatwas in error; that is, the ith column of the matrix is i written as abinary number, and the syndrome immediately tells us which bit is inerror. If the received word is (101011), compute the syndrome. In whichbit did the error occur in this case, and what codeword was originallytransmitted?(c) Give a binary matrix H for the Hamming code with six informationpositions and four check positions. What are the check positions andwhat are the information positions? Encode the messages (101101) and(001001). Decode the received words (0010000101) and (0000101100).What are the possible syndromes for this code?(d) What is the number of check bits and the number of information bitsin an (m, n)-block Hamming code? Give both an upper and a lowerbound on the number of information bits in terms of the number ofcheck bits. Hamming codes having the maximum possible number ofinformation bits with k check bits are called perfect. Every possiblesyndrome except 0 occurs as a column. If the number of informationbits is less than the maximum, then the code is called shortened. Inthis case, give an example showing that some syndromes can representmultiple errors.Programming ExercisesWrite a program to implement a (16, 12)-linear code. Your program should beable to encode and decode messages using coset decoding. Once your program iswritten, write a program to simulate a binary symmetric channel with transmissionnoise. Compare the results of your simulation with the theoretically predicted errorprobability.References and Suggested Readings[1] Blake, I. F. “Codes and Designs,” Mathematics Magazine 52 (1979), 81–95.[2] Hill, R. A First Course in Coding Theory. Oxford University Press, Oxford,1986.[3] Levinson, N. “Coding Theory: A Counterexample to G. H. Hardy’s Conceptionof Applied Mathematics,” American Mathematical Monthly 77 (1970),249–58.[4] Lidl, R. and Pilz, G. Applied Abstract Algebra. Springer-Verlag, New York,1984.
EXERCISES 137[5] MacWilliams, F. J. and Sloane, N. J. A. The Theory of Error-CorrectingCodes. North Holland, Amsterdam, 1977.[6] Roman, S. Coding and Information Theory. Springer-Verlag, New York,1992.[7] Shannon, C. E. “A Mathematical Theory of Communication,” Bell SystemTechnical Journal 27 (1948), 379–423, 623–56.[8] Thompson, T. M. From Error-Correcting Codes through Sphere Packing toSimple Groups. Carus Monograph Series, No. 21. Mathematical Associationof America, Washington, DC, 1983.[9] van Lint, J. H. Introduction to Coding Theory. Springer-Verlag, New York,1982.
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Abstract AlgebraTheory and Applicat
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PrefaceThis text is intended for a
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PREFACEixappears at the end of each
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CONTENTSxi6 Introduction to Cryptog
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0PreliminariesA certain amount of m
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0.1 A SHORT NOTE ON PROOFS 3ax 2 +
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0.2 SETS AND EQUIVALENCE RELATIONS
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0.2 SETS AND EQUIVALENCE RELATIONS
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10 CHAPTER 0 PRELIMINARIESnot all f
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12 CHAPTER 0 PRELIMINARIESThis is a
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14 CHAPTER 0 PRELIMINARIESgiven by
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16 CHAPTER 0 PRELIMINARIESandB =the
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18 CHAPTER 0 PRELIMINARIESIf we con
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20 CHAPTER 0 PRELIMINARIES(e) If g
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1The IntegersThe integers are the b
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24 CHAPTER 1 THE INTEGERSwhere a an
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26 CHAPTER 1 THE INTEGERSEvery good
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28 CHAPTER 1 THE INTEGERSThe Euclid
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30 CHAPTER 1 THE INTEGERSTheorem 1.
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32 CHAPTER 1 THE INTEGERS9. Use ind
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34 CHAPTER 1 THE INTEGERSProgrammin
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36 CHAPTER 2 GROUPSZ n . Consider t
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38 CHAPTER 2 GROUPSSymmetriesA symm
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40 CHAPTER 2 GROUPSTable 2.2. Symme
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42 CHAPTER 2 GROUPSand 4. We define
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44 CHAPTER 2 GROUPSis the inverse o
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46 CHAPTER 2 GROUPS1. mg + ng = (m
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48 CHAPTER 2 GROUPSnecessarily obta
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50 CHAPTER 2 GROUPS3. Write out Cay
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52 CHAPTER 2 GROUPS32. Find all the
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54 CHAPTER 2 GROUPS0 50000 30042 6F
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3Cyclic GroupsThe groups Z and Z n
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58 CHAPTER 3 CYCLIC GROUPS✦ ❛
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60 CHAPTER 3 CYCLIC GROUPS3.2 The G
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62 CHAPTER 3 CYCLIC GROUPSanda = r
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64 CHAPTER 3 CYCLIC GROUPSTheorem 3
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66 CHAPTER 3 CYCLIC GROUPSk multipl
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68 CHAPTER 3 CYCLIC GROUPS4. Find t
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70 CHAPTER 3 CYCLIC GROUPS27. If g
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4Permutation GroupsPermutation grou
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74 CHAPTER 4 PERMUTATION GROUPSthis
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76 CHAPTER 4 PERMUTATION GROUPSExam
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78 CHAPTER 4 PERMUTATION GROUPSExam
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80 CHAPTER 4 PERMUTATION GROUPSProo
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82 CHAPTER 4 PERMUTATION GROUPSresu
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84 CHAPTER 4 PERMUTATION GROUPSand
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Page 98 and 99: 90 CHAPTER 5 COSETS AND LAGRANGE’
Page 106 and 107: 98 CHAPTER 6 INTRODUCTION TO CRYPTO
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Page 116 and 117: 7Algebraic Coding TheoryCoding theo
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Page 146 and 147: 8IsomorphismsMany groups may appear
Page 148 and 149: 140 CHAPTER 8 ISOMORPHISMSab ≠ ba
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Page 154 and 155: 146 CHAPTER 8 ISOMORPHISMSTherefore
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Page 160 and 161: 9Homomorphisms and FactorGroupsIf H
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Page 178 and 179: 10Matrix Groups andSymmetryWhen Fel
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186 CHAPTER 10 MATRIX GROUPS AND SY
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188 CHAPTER 10 MATRIX GROUPS AND SY
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11The Structure of GroupsThe ultima
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192 CHAPTER 11 THE STRUCTURE OF GRO
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204 CHAPTER 12 GROUP ACTIONSThe ele
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206 CHAPTER 12 GROUP ACTIONSand the
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208 CHAPTER 12 GROUP ACTIONSExample
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210 CHAPTER 12 GROUP ACTIONSLemma 1
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212 CHAPTER 12 GROUP ACTIONSinduces
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214 CHAPTER 12 GROUP ACTIONSSwitchi
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216 CHAPTER 12 GROUP ACTIONSTable 1
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218 CHAPTER 12 GROUP ACTIONS10. Fin
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13The Sylow TheoremsWe already know
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222 CHAPTER 13 THE SYLOW THEOREMSHe
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224 CHAPTER 13 THE SYLOW THEOREMSpo
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226 CHAPTER 13 THE SYLOW THEOREMSbe
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228 CHAPTER 13 THE SYLOW THEOREMSSu
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230 CHAPTER 13 THE SYLOW THEOREMS(e
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14RingsUp to this point we have stu
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234 CHAPTER 14 RINGSExample 3. We c
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236 CHAPTER 14 RINGS3. (−a)(−b)
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238 CHAPTER 14 RINGSProposition 14.
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240 CHAPTER 14 RINGSa ring homomorp
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242 CHAPTER 14 RINGSTheorem 14.9 Le
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244 CHAPTER 14 RINGSTheorem 14.15 L
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246 CHAPTER 14 RINGSthe Senate is n
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248 CHAPTER 14 RINGShas a solution.
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250 CHAPTER 14 RINGSMultiplying the
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252 CHAPTER 14 RINGS11. Prove that
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254 CHAPTER 14 RINGS34. Let R be a
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15PolynomialsMost people are fairly
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258 CHAPTER 15 POLYNOMIALSExample 1
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260 CHAPTER 15 POLYNOMIALSProof. Su
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262 CHAPTER 15 POLYNOMIALSm ≥ n.
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264 CHAPTER 15 POLYNOMIALSPropositi
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268 CHAPTER 15 POLYNOMIALSwhere eac
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272 CHAPTER 15 POLYNOMIALS(c) p(x)
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274 CHAPTER 15 POLYNOMIALSAdditiona
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276 CHAPTER 15 POLYNOMIALS(a) x 4
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278 CHAPTER 16 INTEGRAL DOMAINSelem
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280 CHAPTER 16 INTEGRAL DOMAINSIt i
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282 CHAPTER 16 INTEGRAL DOMAINSLet
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284 CHAPTER 16 INTEGRAL DOMAINSCons
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286 CHAPTER 16 INTEGRAL DOMAINSEucl
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288 CHAPTER 16 INTEGRAL DOMAINSin D
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290 CHAPTER 16 INTEGRAL DOMAINSCoro
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292 CHAPTER 16 INTEGRAL DOMAINS7. L
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17Lattices and BooleanAlgebrasThe a
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296 CHAPTER 17 LATTICES AND BOOLEAN
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18Vector SpacesIn a physical system
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314 CHAPTER 18 VECTOR SPACES5. −(
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316 CHAPTER 18 VECTOR SPACESProposi
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318 CHAPTER 18 VECTOR SPACES3. If S
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320 CHAPTER 18 VECTOR SPACES(d) Let
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19FieldsIt is natural to ask whethe
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324 CHAPTER 19 FIELDS· 0 1 α 1 +
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326 CHAPTER 19 FIELDSExample 5. We
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328 CHAPTER 19 FIELDSfor b i and c
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330 CHAPTER 19 FIELDSLetu = ∑ i,j
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332 CHAPTER 19 FIELDSwhere each fie
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334 CHAPTER 19 FIELDSLet F be a fie
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336 CHAPTER 19 FIELDSTheorem 19.19
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338 CHAPTER 19 FIELDSlength αβ. A
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340 CHAPTER 19 FIELDSbe the equatio
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342 CHAPTER 19 FIELDScosine,4α 3
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344 CHAPTER 19 FIELDS11. Let p(x) b
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20Finite FieldsFinite fields appear
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348 CHAPTER 20 FINITE FIELDSextensi
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350 CHAPTER 20 FINITE FIELDSGF(p 24
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352 CHAPTER 20 FINITE FIELDSMessage
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354 CHAPTER 20 FINITE FIELDSThe add
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356 CHAPTER 20 FINITE FIELDSExample
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358 CHAPTER 20 FINITE FIELDSand g(x
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360 CHAPTER 20 FINITE FIELDS(3) ⇒
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362 CHAPTER 20 FINITE FIELDS22. Sho
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21Galois TheoryA classic problem of
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366 CHAPTER 21 GALOIS THEORYthe Gal
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368 CHAPTER 21 GALOIS THEORYof G(E/
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370 CHAPTER 21 GALOIS THEORYin K. A
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372 CHAPTER 21 GALOIS THEORYThe pro
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374 CHAPTER 21 GALOIS THEORY{id, σ
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376 CHAPTER 21 GALOIS THEORYThe ker
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378 CHAPTER 21 GALOIS THEORYfor res
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380 CHAPTER 21 GALOIS THEORYis a no
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382 CHAPTER 21 GALOIS THEORYand onc
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384 CHAPTER 21 GALOIS THEORY(a) x 5
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386 CHAPTER 21 GALOIS THEORY[3] Fra
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388 NOTATIONSymbol Description Page
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390 NOTATIONSymbol Description Page
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392 HINTS AND SOLUTIONSConversely,
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394 HINTS AND SOLUTIONS4. (a)(c)( 1
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396 HINTS AND SOLUTIONS7. (a) (0000
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398 HINTS AND SOLUTIONS9. For any h
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400 HINTS AND SOLUTIONS2. The four
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402 HINTS AND SOLUTIONSChapter 17.
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404 HINTS AND SOLUTIONSChapter 19.
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GNU Free Documentation LicenseVersi
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408 GFDL LICENSEappear in the title
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410 GFDL LICENSEH. Include an unalt
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412 GFDL LICENSEply to the other wo
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IndexG-equivalent, 205G-set, 203nth
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416 INDEXEquivalence relation, 15Eu
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418 INDEXKlein, Felix, 46, 170, 245
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420 INDEXnormalizer of, 222proper,
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