On Generalized Hamming Weights and the Covering Radius of Linear Codes

  • H. Janwa
  • A. K. Lal
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4851)


We prove an upper bound on the covering radius of linear codes over \({\mathbb{F}_q}\)in terms of their generalized Hamming weights. We show that this bound is strengthened if we know that the codes satisfy the chain condition or a partial chain condition. We show that this bound improves all prior bounds. Necessary conditions for equality are also presented.

Several applications of our bound are presented. We give tables of improved bounds on the covering radius of many cyclic codes using their generalized Hamming weights. We show that most cyclic codes of length ≤ 39 satisfy the chain condition or partial chain condition up to level 5. We use these results to derive tighter bounds on the covering radius of cyclic codes.


Generalized Hamming weights covering radius Griesmer bound optimal codes cyclic codes chain condition generalized Griesmer bound 


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  1. 1.
    Berlekamp, E.R., McEliece, R.J., van Tilborg, H.C.A.: On the Inherent Intractability of Some Coding Problems. IEEE Trans. Inform. Theory 24(3), 384–386 (1996)CrossRefGoogle Scholar
  2. 2.
    Cohen, G., Honkala, I., Litsyn, S., Lobstein, A.: Covering Codes. In: Sakata, S. (ed.) AAECC-8. LNCS, vol. 508, pp. 173–239. Springer, Heidelberg (1991)Google Scholar
  3. 3.
    Dougherty, R., Janwa, H.: Covering Radius Computations for Binary Cyclic Codes. Math. Comp. 57(195), 415–434 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Encheva, S., Kløve, T.: Codes Satisfying the Chain Condition. IEEE Trans. Inform. Theory 40(1), 175–180 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Forney, G.D.: Dimension/Length Profiles and Trellis Complexity of Linear Block Codes. IEEE Trans. Inform. Theory 40(6), 1741–1752 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Guruswami, V.: List Decoding From Erasures: Bounds and Code Constructions. IEEE Trans. Inform. Theory 49(11), 2826–2833 (2003)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Heijnen, P., Pellikaan, R.: Generalized Hamming Weights of q-ARY Reed-Muller Codes. IEEE Trans. Inform. Theory 44(1), 181–196 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Helleseth, T., Kløve, T., Ytrehus, Ø.: Codes, Weight Hierarchies, and Chains. In: 1992 ICCS/ISITA, Singapore, pp. 608–612 (1992)Google Scholar
  9. 9.
    Helleseth, T., Kløve, T., Ytrehus, Ø.: Generalized Hamming Weights of Linear Codes. IEEE Trans. Inform. Theory 38(3), 1133–1140 (1992)Google Scholar
  10. 10.
    Helleseth, T., Kløve, T., Levenshtein, V.I., Ytrehus, Ø.: Bounds on the Minimum Support Weights. IEEE Trans. Inform. Theory 41(2), 432–440 (1995)Google Scholar
  11. 11.
    Helleseth, T. , Kløve, T. , Levenshtein, V. I., Ytrehus, Ø.: Excess Sequences of Codes and the Chain Condition. In: Reports in Informatics, no. 65, Department of Informatics, University of Bergen (1993)Google Scholar
  12. 12.
    Janwa, H.: On the Optimality and Covering Radii of Some Algebraic Geometric Codes. In: Workshop on Coding Theory, IMA, University of Minnesota (1988)Google Scholar
  13. 13.
    Janwa, H.: Some New Upper Bounds on the Covering Radius of Binary Linear Codes. IEEE Trans. Inform. Theory 35, 110–122 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Janwa, H.: On the Covering Radii of q-ary Codes. In: 1990 ISIT, San DiegoGoogle Scholar
  15. 15.
    Janwa, H.: Some Optimal Codes From Algebraic Geometry and Their Covering Radii. Europ. J. Combinatorics 11, 249–266 (1990)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Janwa, H.: On the Covering Radii of AG Codes (preprint, 2007)Google Scholar
  17. 17.
    Janwa, H., Lal, A.K.: On the Generalized Hamming Weights of Cyclic Codes. IEEE Trans. Inform. Theory 43(1), 299–308 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Janwa, H., Lal, A.K.: Bounds on the Covering Radii of Codes in Terms of Their Generalized Hamming Weights. MRI (preprint, 1997)Google Scholar
  19. 19.
    Janwa, H., Lal, A.K.: Upper Bounds on the Covering Radii of Some Important Classes of Codes Using Their Generalized Hamming Weights (preprint, 2007)Google Scholar
  20. 20.
    Janwa, H., Mattson Jr., H.F.: Some Upper Bounds on the Covering Radii of Linear Codes over F q and Their Applications. Designs, Codes and Cryptography 18(1-3), 163–181 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Kløve, T.: Minimum Support Weights of Binary Codes. IEEE Trans. Inform. Theory 39(2), 648–654 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    MacWilliaims, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977)Google Scholar
  23. 23.
    Mattson Jr., H.F.: An Improved Upper Bound on Covering Radius. In: Poli, A. (ed.) AAECC-2. LNCS, vol. 228, pp. 90–106. Springer, Heidelberg (1986)Google Scholar
  24. 24.
    Ozarow, L.H., Wyner, A.D.: Wire-Tap Channel-II. AT & T Bell Labs Tech J. 63, 2135–2157 (1984)zbMATHGoogle Scholar
  25. 25.
    Pless, V.S., Huffman, W.C., Brualdi, R.A.: An Introduction to Algebraic Codes. In: Pless, V.S., Huffman, W.C. (eds.) Handbook of Coding Theory, pp. 3–139. Elsevier, Amsterdam (1998)Google Scholar
  26. 26.
    Wei, V.K.: Generalized Hamming Weights for Linear Codes. IEEE Trans. Inform. Theory 37(5), 1412–1418 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Wei, V.K., Yang, K.: The Feneralized Hamming Weights for Product Codes. IEEE Trans. Inform. Theory 39(5), 1709–1713 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Yang, K., Kumar, P.V., Stichtenoth, H.: On the Weight Hierarchy of Geometric Goppa Codes. IEEE Trans. Inform. Theory 40(3), 913–920 (1994)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • H. Janwa
    • 1
  • A. K. Lal
    • 2
  1. 1.Department of Mathematics and Computer Science, University of Puerto Rico (UPR), Rio Piedras Campus, P.O. Box: 23355, San Juan, PR 00931 - 3355 
  2. 2.Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, 208016India

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