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# A Complete Linear Programming Hierarchy for Linear Codes

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LIPIcs.ITCS.2022.51.pdf
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## Acknowledgements

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

## Cite As

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

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## 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.
3. P. Delsarte and V. I. Levenshtein. Association schemes and coding theory. IEEE Transactions on Information Theory, 44(6):2477-2504, 1998.
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.
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.
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.
8. Monique Laurent. Strengthened semidefinite programming bounds for codes. Mathematical Programming, 109:1436-4646, 2007.
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.
10. Jessie MacWilliams. A theorem on the distribution of weights in a systematic code†. Bell System Technical Journal, 42(1):79-94, 1963.
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.
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.
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.
14. Michael Navon and Alex Samorodnitsky. Linear programming bounds for codes via a covering argument. Discrete Comput. Geom., 41(2):199-207, March 2009.
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.
17. A. Schrijver. New code upper bounds from the Terwilliger algebra and semidefinite programming. IEEE Transactions on Information Theory, 51(8):2859-2866, 2005.
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.
20. R.R. Varshamov. Estimate of the number of signals in error correcting codes. Doklady Akademii Nauk SSSR, 117:739-741, 1957.
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