Semidefinite programming bounds for error-correcting codes

Vallentin, Frank

doi:10.48550/arxiv.1902.01253

Cited by 2 publications

(3 citation statements)

References 52 publications

(58 reference statements)

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“…In Delsarte's approach, only the distance between pairs of points is taken into account in the optimization. For this reason, Delsarte's approach is classified as a 2-point bound [Val19]. Nonetheless, there is no reason to restrict oneself to just 2-point interactions.…”

Section: More On Related Workmentioning

confidence: 99%

“…It turns out that KrawtchoukLP is a symmetrization of FourierLP (and likewise for the programs KrawtchoukLP Lin and FourierLP Lin ). We will briefly describe the technique of symmetrizing convex programs, which is also described in the survey article by Vallentin [Val19]. The proof that KrawtchoukLP and FourierLP are equivalent continues at Proposition 4.5.…”

Section: The Hierarchy As Checking Non-negativity Of Fourier Coeffici...mentioning

confidence: 99%

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A Complete Linear Programming Hierarchy for Linear Codes

Coregliano¹,

Jeronimo²,

Jones³

2021

Preprint

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A longstanding open problem in coding theory is to determine the best (asymptotic) rate R 2 (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 R 2 (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: i. It is a natural generalization of the Delsarte LPs used in the first MRRW bound.ii. It is a hierarchy of linear programs rather than semi-definite programs potentially making it more amenable to theoretical analysis.iii. It is complete in the sense that the optimum code size can be retrieved from level O(n 2 ).iv.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.

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Section: More On Related Workmentioning

confidence: 99%

Section: The Hierarchy As Checking Non-negativity Of Fourier Coeffici...mentioning

confidence: 99%

A Complete Linear Programming Hierarchy for Linear Codes

Coregliano¹,

Jeronimo²,

Jones³

2021

Preprint

View full text Add to dashboard Cite

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“…This body of work computes upper bounds for the largest distance d codes in some setting through semidefinite programming. See [Val19] for a more complete overview of this area. We find it natural to then wonder whether graphical designs are the same maximum stable sets, possibly in a distance graph, or whether optimal designs are stable sets at all.…”

Section: Definition 51 ([Gol20]mentioning

confidence: 99%

Codes, Cubes, and Graphical Designs

Babecki¹

2020

Preprint

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Graphical designs are an extension of spherical designs to functions on graphs. We connect linear codes to graphical designs on cube graphs, and show that the Hamming code in particular is a highly effective graphical design. We show that even in highly structured graphs, graphical designs are distinct from the related concepts of extremal designs, maximum stable sets in distance graphs, and t-designs on association schemes.

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Semidefinite programming bounds for error-correcting codes

Cited by 2 publications

References 52 publications

A Complete Linear Programming Hierarchy for Linear Codes

A Complete Linear Programming Hierarchy for Linear Codes

Codes, Cubes, and Graphical Designs

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