Error Correction I: Distance Concepts and Bounds

  • Solomon W. Golomb
  • Robert E. Peile
  • Robert A. Scholtz
Part of the Applications of Communications Theory book series (ACTH)


Agent 00111 was worried! He was firmly established on foreign soil and had been communicating with Whitehall regularly in code on the back of postage stamps. The system worked beautifully until the local post office changed clerks. Using lots of ink and a heavy hand, the new clerk had managed to render illegible parts of 13 consecutive posted messages by canceling the stamps! While the messages were not completely obliterated, some symbols were impossible to read.


Error Correction Word Length Channel Matrix Code Word Perfect Code 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. L. A. Bassalygo. 1965. “New Upper Bounds for Error-Correcting Codes.” Probl. Peredachi. Inf. 1: 41–45.MathSciNetzbMATHGoogle Scholar
  2. E. R. Berlekamp, ed. 1974. Key Papers in the Development of Coding Theory. IEEE Press, New York.zbMATHGoogle Scholar
  3. H. O. Burton and D. D. Sullivan, 1972. “Error and Error Control.” Proc. IEEE 60: 1293–1301.CrossRefGoogle Scholar
  4. J. H. Conway and N. J. A. Sloane. 1988. Sphere Packings, Lattices and Groups. Springer-Verlag, New York.zbMATHGoogle Scholar
  5. P. Delsarte. 1973. An Algebraic Approach to the Association Schemes of Coding Theory. Philips Research Reports Supplements No. 10, Eindhoven.zbMATHGoogle Scholar
  6. W. Feller. 1950. An Introduction to Probability Theory and its Applications Vol. 1: Wiley, New York.Google Scholar
  7. G. D. Forney, Jr. 1972. “Maximum-Likelihood Sequence Estimation of Digital Sequences in the Presence of Intersymbol Interference.” IEEE Trans. Inform. Theory 18: 363–78.MathSciNetzbMATHCrossRefGoogle Scholar
  8. G. D. Forney, Jr. 1973. “The Viterbi Algorithm.” Proc. IEEE 61: 268–78.MathSciNetCrossRefGoogle Scholar
  9. E. N. Gilbert. 1952. “A Comparison of Signalling Alphabets.” Bell Sys. Tech. J. 31: 504–22.Google Scholar
  10. M. J. E. Golay. 1949. “Notes on Digital Coding.” Proc. IRE 37: 657.Google Scholar
  11. S. W. Golomb and L. R. Welch. 1968. “Algebraic Coding in the Lee Metric.” In Error-Correcting Codes, edited by H. B. Mann. Wiley, New York.Google Scholar
  12. R. W. Hamming. 1950. “Error-Detecting and Error-Correcting Codes.” Bell Sys. Tech. J. Vol. 29: 147–60.MathSciNetGoogle Scholar
  13. S. M. Johnson. 1971. “On Upper Bounds for Unrestricted Binary-Error-Correcting Codes.” IEEE Trans. Inform. Theory 17: 466–78.zbMATHCrossRefGoogle Scholar
  14. G. A. Kabatiansky and V. I. Levenshtein. 1978. “Bounds for Packings on a Sphere and in Space.” PPI 14: pp. 3–25.Google Scholar
  15. J. H. van Lint. 1971. Coding Theory. Lecture Notes in Mathematics 201, Springer-Verlag, New York.zbMATHCrossRefGoogle Scholar
  16. F. J. MacWilliams. 1963. “A Theorem on the Distribution of Weights in a Systematic Code.” Bell Sys. Tech. J. 42: 79–84.MathSciNetGoogle Scholar
  17. F. J. MacWilliams and N. J. A. Sloane. 1977. The Theory of Error-Correcting Codes. North-Holland, Amsterdam.zbMATHGoogle Scholar
  18. R. J. McEliece, H. Rumsey, Jr., and L. R. Welch. 1974. “A Low-Rate Improvement on the Elias Bound.” IEEE Trans. Inform. Theory 20: 676–78.MathSciNetzbMATHCrossRefGoogle Scholar
  19. R. J. McEliece, E. R. Rodemich, H. Rumsey, Jr., and L. R. Welch. 1977. “New Upper Bounds on the Rate of a Code via the Delsarte-MacWilliams Inequalities.” IEEE Trans. Inform. Theory 23: 157–66.MathSciNetzbMATHCrossRefGoogle Scholar
  20. V. Pless. 1963. “Power Moment Identities on the Weight Distributions in Error-Correcting Codes.” Inf . Contr. 6: 147–62.MathSciNetzbMATHCrossRefGoogle Scholar
  21. M. Plotkin. 1960. “Binary Codes with Specified Minimum Distance.” IRE Trans. Inform. Theory 6: 445–50.MathSciNetCrossRefGoogle Scholar
  22. N. J. A. Sloane. 1972. “A Survey of Constructive Coding Theory and a Table of Binary Codes of Highest Known Rate.” Discrete Math. 3: 265–94.MathSciNetzbMATHCrossRefGoogle Scholar
  23. J. J. Stiffler, ed. 1971. “Special Issue on Error-Correcting Codes.” IEEE Trans. Commun. Technol. 19, no. 5, part II.Google Scholar
  24. A. Tietäväinen. 1973. “On the Nonexistence of Perfect Codes over Finite Fields.” SIAM J. Appl. Math. 24: 88–96.MathSciNetzbMATHCrossRefGoogle Scholar
  25. R. R. Varshamov. 1957. “Estimate of the Number of Signals in Error-Correcting Codes.” Dokl. Akad. Nauk SSSR 117: 739–41.MathSciNetzbMATHGoogle Scholar
  26. T. Verhoeff. 1987. “An Updated Table of Minimum Distance Bounds for Binary Linear Codes.” IEEE Trans. on Information Theory 33: 665–680.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • Solomon W. Golomb
    • 1
  • Robert E. Peile
    • 2
  • Robert A. Scholtz
    • 3
  1. 1.Departments of Electrical Engineering and MathematicsUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Racal Research, LimitedReading, BerkshireUK
  3. 3.Department of Electrical EngineeringUniversity of Southern CaliforniaLos AngelesUSA

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