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Bounds for Covering Codes over Large Alphabets

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Abstract

Let K q (n,R) denote the minimum number of codewords in any q-ary code of length n and covering radius R. We collect lower and upper bounds for K q (n,R) where 6 ≤ q ≤ 21 and R ≤ 3. For q ≤ 10, we consider lengths n ≤ 10, and for q ≥ 11, we consider n ≤ 8. This extends earlier results, which have been tabulated for 2 ≤ q ≤ 5. We survey known bounds and obtain some new results as well, also for s-surjective codes, which are closely related to covering codes and utilized in some of the constructions.

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References

  • R. Bertolo P.R.J. Ögård W.D Weakley (2004) ArticleTitleAn updated table of binary/ternary mixed covering codes Journal of Combinatorial Designs. 12 157–176 Occurrence Handle10.1002/jcd.20008

    Article  Google Scholar 

  • M.C. Bhandari C. Durairajan (1996) ArticleTitleA note on bounds for q-ary covering codes IEEE Transactions Information Theory. 42 1640–1642 Occurrence Handle10.1109/18.532916

    Article  Google Scholar 

  • A. Blokhuis C.W.H. Lam (1984) ArticleTitleMore coverings by rook domains Journal of Combinatorial Theory Series A. 36 240–244 Occurrence Handle10.1016/0097-3165(84)90010-4

    Article  Google Scholar 

  • M. Chateauneuf DL. Kreher (2002) ArticleTitleOn the state of strength-three covering arrays Journal of Combinatorial Designs. 10 217–238 Occurrence Handle10.1002/jcd.10002

    Article  Google Scholar 

  • W. Chen IS. Honkala (1990) ArticleTitleLower bounds for q-ary covering codes IEEE Transactions on Information Theory. 36 664–671 Occurrence Handle10.1109/18.54887

    Article  Google Scholar 

  • G. Cohen I. Honkala S. Litsyn A Lobstein (1997) Covering Codes North-Holland Amsterdam

    Google Scholar 

  • Cohen MB.,Colbourn C.J., Ling A.C.H. Constructing strength three covering arrays with augmented annealing, Discrete Mathematics, to appear

  • A.A. Davydov P.R.J. Östergård (2000) ArticleTitleOn saturating sets in small projective geometries European Journal of Combinatorics. 21 563–570 Occurrence Handle10.1006/eujc.1999.0373

    Article  Google Scholar 

  • S.W. Golomb EC. Posner (1964) ArticleTitleRook domains, Latin squares, affine planes, and error-distributing codes IEEE Transactions on Information Theory. 10 196–208 Occurrence Handle10.1109/TIT.1964.1053680

    Article  Google Scholar 

  • W. Haas (2000) ArticleTitleLower bounds for q-ary codes of covering radius one Discrete Mathematics. 219 97–106 Occurrence Handle10.1016/S0012-365X(99)00338-6

    Article  Google Scholar 

  • J.W.P. Hirschfeld (1998) Projective Geometries over Finite Fields EditionNumber2 Clarendon Oxford

    Google Scholar 

  • IS. Honkala (1991) ArticleTitleModified bounds for covering codes IEEE Transactions on Information Theory. 37 351–365 Occurrence Handle10.1109/18.75253

    Article  Google Scholar 

  • IS. Honkala (1992) ArticleTitleA Graham–Sloane type construction for s-surjective matrices Journal of Algebraic Combinatorics. 1 347–351 Occurrence Handle10.1023/A:1022490600755

    Article  Google Scholar 

  • J.G. Kalbfleisch RG. Stanton (1969) ArticleTitleA combinatorial theorem of matching Journal of the London Mathematical Society. 44 60–64

    Google Scholar 

  • JG. Kalbfleisch R.G. Stanton JD. Horton (1971) ArticleTitleOn covering sets and error-correcting codes Journal of Combinatorial Theory Series A. 11 233–250 Occurrence Handle10.1016/0097-3165(71)90051-3

    Article  Google Scholar 

  • G. Kéri P.R.J. Östergård (2003) ArticleTitleOn the covering radius of small codes Studia Scientiarum Mathematicarum Hungarica. 40 243–256

    Google Scholar 

  • K. Meagher B. Stevens (2005) ArticleTitleGroup construction of covering arrays Journal of Combinatorial Designs. 13 70–77 Occurrence Handle10.1002/jcd.20035

    Article  Google Scholar 

  • P.R.J. Östergård (1994) Construction methods for mixed covering codes M. Gyllenberg LE. Persson (Eds) Analysis, Algebra, and Computers in Mathematical Research: Proc. 21st Nordic Congress of Mathematicians. Marcel Dekker New York 387–407

    Google Scholar 

  • P.R.J. Östergård H.O. Hämäläinen (1997) ArticleTitleA new table of binary/ternary mixed covering codes Designs Codes Cryptography. 11 151–178 Occurrence Handle10.1023/A:1008228721072

    Article  Google Scholar 

  • P.R.J. Östergård MK. Kaikkonen (1998) ArticleTitleNew upper bounds for binary covering codes Discrete Mathematics. 178 165–179 Occurrence Handle10.1016/S0012-365X(97)81825-0

    Article  Google Scholar 

  • P.R.J. Östergård A. Wassermann (2002) ArticleTitleA new lower bound for the football pool problem for six matches Journal of Combinatorial Theory Series A. 99 175–179 Occurrence Handle10.1006/jcta.2002.3260

    Article  Google Scholar 

  • ER. Rodemich (1970) ArticleTitleCoverings by rook domains Journal of Combinatorial Theory. 9 117–128

    Google Scholar 

  • RG. Stanton J.D. Horton J.G Kalbfleisch (1969) ArticleTitleCovering theorems for vectors with special reference to the case of four and five components Journal of London Mathematical Society. 1 493–499

    Google Scholar 

  • O. Taussky J. Todd (1948) ArticleTitleCovering theorems for groups Annales Society Polon Mathematics. 21 303–305

    Google Scholar 

  • G.J.M. Wee Particlevan (1991) ArticleTitleBounds on packings and coverings by spheres in q-ary and mixed Hamming spaces Journal of Combinatorial Theory Series A. 57 117–129 Occurrence Handle10.1016/0097-3165(91)90010-E

    Article  Google Scholar 

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Authors and Affiliations

  1. Computer and Automation Research Institute, Hungarian Academy of Sciences, H-1111, Budapest Kende u. 13-17, Hungary

    Gerzson Kéri

  2. Department of Electrical and Communications Engineering, Helsinki University of Technology, P.O. Box 3000, 02015, HUT, Finland

    Patric R. J. Östergård

Authors
  1. Gerzson Kéri
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  2. Patric R. J. Östergård
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Correspondence to Gerzson Kéri.

Additional information

Communicated by: P. Charpin

AMS Classification: 94B75, 94B25, 94B65

Gerzson Kéri - Supported in part by the Hungarian National Research Fund, Grant No. OTKA-T029572.

Patric R. J. Östergård - Supported in part by the Academy of Finland, Grants No. 100500 and No. 202315.

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Kéri, G., Östergård, P.R.J. Bounds for Covering Codes over Large Alphabets. Des Codes Crypt 37, 45–60 (2005). https://doi.org/10.1007/s10623-004-3804-8

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  • DOI: https://doi.org/10.1007/s10623-004-3804-8

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