Document Open Access Logo

A Complete Linear Programming Hierarchy for Linear Codes

Authors Leonardo Nagami Coregliano, Fernando Granha Jeronimo, Chris Jones



PDF
Thumbnail PDF

File

LIPIcs.ITCS.2022.51.pdf
  • Filesize: 0.78 MB
  • 22 pages

Document Identifiers

Author Details

Leonardo Nagami Coregliano
  • Institute for Advanced Study, Princeton, NJ, USA
Fernando Granha Jeronimo
  • Institute for Advanced Study, Princeton, NJ, USA
Chris Jones
  • University of Chicago, IL, USA

Acknowledgements

We would like to thank the anonymous reviewers for their comments and helpful feedback.

Cite AsGet BibTex

Leonardo Nagami Coregliano, Fernando Granha Jeronimo, and Chris Jones. A Complete Linear Programming Hierarchy for Linear Codes. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 51:1-51:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ITCS.2022.51

Abstract

A longstanding open problem in coding theory is to determine the best (asymptotic) rate R₂(δ) of binary codes with minimum constant (relative) distance δ. An existential lower bound was given by Gilbert and Varshamov in the 1950s. On the impossibility side, in the 1970s McEliece, Rodemich, Rumsey and Welch (MRRW) proved an upper bound by analyzing Delsarte’s linear programs. To date these results remain the best known lower and upper bounds on R₂(δ) with no improvement even for the important class of linear codes. Asymptotically, these bounds differ by an exponential factor in the blocklength. In this work, we introduce a new hierarchy of linear programs (LPs) that converges to the true size A^{Lin}₂(n,d) of an optimum linear binary code (in fact, over any finite field) of a given blocklength n and distance d. This hierarchy has several notable features: 1) It is a natural generalization of the Delsarte LPs used in the first MRRW bound. 2) It is a hierarchy of linear programs rather than semi-definite programs potentially making it more amenable to theoretical analysis. 3) It is complete in the sense that the optimum code size can be retrieved from level O(n²). 4) It provides an answer in the form of a hierarchy (in larger dimensional spaces) to the question of how to cut Delsarte’s LP polytopes to approximate the true size of linear codes. We obtain our hierarchy by generalizing the Krawtchouk polynomials and MacWilliams inequalities to a suitable "higher-order" version taking into account interactions of 𝓁 words. Our method also generalizes to translation schemes under mild assumptions.

Subject Classification

ACM Subject Classification
  • Theory of computation
Keywords
  • Coding theory
  • code bounds
  • convex programming
  • linear programming hierarchy

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Etienne de Klerk, Dmitrii V. Pasechnik, and Alexander Schrijver. Reduction of symmetric semidefinite programs using the regular ∗-representation. Math. Program., 109(2-3, Ser. B):613-624, 2007. URL: https://doi.org/10.1007/s10107-006-0039-7.
  2. P. Delsarte. An Algebraic Approach to the Association Schemes of Coding Theory. Philips Journal of Research / Supplement. N.V. Philips' Gloeilampenfabrieken, 1973. Google Scholar
  3. P. Delsarte and V. I. Levenshtein. Association schemes and coding theory. IEEE Transactions on Information Theory, 44(6):2477-2504, 1998. Google Scholar
  4. Joel Friedman and Jean-Pierre Tillich. Generalized Alon-Boppana theorems and error-correcting codes. SIAM J. Discret. Math., 19(3):700-718, July 2005. Google Scholar
  5. D. C. Gijswijt, H. D. Mittelmann, and A. Schrijver. Semidefinite code bounds based on quadruple distances. IEEE Transactions on Information Theory, 58(5):2697-2705, 2012. Google Scholar
  6. Dion Gijswijt. Block diagonalization for algebra’s associated with block codes, 2009. URL: http://arxiv.org/abs/0910.4515.
  7. E.N. Gilbert. A comparison of signalling alphabets. Bell System Technical Journal, 31:504-522, 1952. Google Scholar
  8. Monique Laurent. Strengthened semidefinite programming bounds for codes. Mathematical Programming, 109:1436-4646, 2007. Google Scholar
  9. Monique Laurent. Sums of squares, moment matrices and optimization over polynomials. In Emerging Applications of Algebraic Geometry (of IMA Volumes in Mathematics and its Applications). Springer, 2009. Google Scholar
  10. Jessie MacWilliams. A theorem on the distribution of weights in a systematic code†. Bell System Technical Journal, 42(1):79-94, 1963. Google Scholar
  11. Mrs. F. J. MacWilliams, N. J. A. Sloane, and J.M. Goethals. The MacWilliams identities for nonlinear codes. The Bell System Technical Journal, 51(4):803-819, 1972. Google Scholar
  12. R. McEliece, E. Rodemich, H. Rumsey, and L. Welch. New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities. IEEE Transactions on Information Theory, 23(2):157-166, 1977. Google Scholar
  13. M. Navon and A. Samorodnitsky. On Delsarte’s linear programming bounds for binary codes. In 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05), pages 327-336, 2005. Google Scholar
  14. Michael Navon and Alex Samorodnitsky. Linear programming bounds for codes via a covering argument. Discrete Comput. Geom., 41(2):199-207, March 2009. Google Scholar
  15. Alex Samorodnitsky. One more proof of the first linear programming bound for binary codes and two conjectures, 2021. URL: http://arxiv.org/abs/2104.14587.
  16. A. Schrijver. A comparison of the Delsarte and Lovász bounds. IEEE Transactions on Information Theory, 25(4):425-429, 1979. Google Scholar
  17. A. Schrijver. New code upper bounds from the Terwilliger algebra and semidefinite programming. IEEE Transactions on Information Theory, 51(8):2859-2866, 2005. Google Scholar
  18. Frank Vallentin. Semidefinite programming bounds for error-correcting codes. CoRR, abs/1902.01253, 2019. URL: http://arxiv.org/abs/1902.01253.
  19. Jacobus H. van Lint. Introduction to Coding Theory. Springer-Verlag, 1999. Google Scholar
  20. R.R. Varshamov. Estimate of the number of signals in error correcting codes. Doklady Akademii Nauk SSSR, 117:739-741, 1957. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail