Locally recoverable codes from planar graphs

Year 2020, Volume: 7 Issue: 1 (Special Issue in Algebraic Coding Theory: New Trends and Its Connections), 35 - 53, 29.02.2020

Kathryn Haymaker Justin O'pella

https://doi.org/10.13069/jacodesmath.645021

Cited By: 1

https://izlik.org/JA69NL24CP

Abstract

In this paper we apply Kadhe and Calderbank's definition of LRCs from convex polyhedra and planar graphs \cite{KAD} to analyze the codes resulting from 3-connected regular and almost regular planar graphs. The resulting edge codes are locally recoverable with availability two. We prove that the minimum distance of planar graph LRCs is equal to the girth of the graph, and we also establish a new bound on the rate of planar graph edge codes. Constructions of regular and almost regular planar graphs are given, and their associated code parameters are determined. In certain cases, the code families meet the rate bound.

Keywords

Error-correction , Local recovery , Planar graphs , Availability , Rate bound

References

• [1] H. J. Broersma, A. J. W. Duijvestijn, F. Göbel, Generating all 3-connected 4-regular planar graphs from the octahedron graph, J. Graph Theor. 17(5) (1993) 613–620.
• [2] P. Gopalan, C. Huang, H. Simitci, S. Yekhanin, On the locality of codeword symbols, IEEE Trans. Inform. Theory 58(11) (2012) 6925–6934.
• [3] M. Hasheminezhad, B. D. McKay, T. Reeves, Recursive generation of simple planar 5-regular graphs and pentangulations, Journal of Graph Algorithms and Applications 15(3) (2011) 417–436.
• [4] S. Kadhe, R. Calderbank, Rate optimal binary linear locally repairable codes with small availability, In 2017 IEEE International Symposium on Information Theory (ISIT) (2017) 166–170.
• [5] S. Kadhe, R. Calderbank, Rate optimal binary linear locally repairable codes with small availability, arXiv preprint, arXiv:1701.02456, 2017.
• [6] M. Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theor. 30(2) (1999) 137–146.
• [7] M. Meringer, Regular planar graphs, available online at http://www.mathe2.uni-bayreuth.de/ markus/reggraphs.html, accessed 2009.
• [8] N. Prakash, V. Lalitha, P. Vijay Kumar, Codes with locality for two erasures, In 2014 IEEE International Symposium on Information Theory (2014) 1962–1966.
• [9] E. F. Schmeichel, S. L. Hakimi, On planar graphical degree sequences, SIAM J. Appl. Math. 32(3)(1977) 598–609.
• [10] E. Steinitz, Vorlesungen {\"u}ber die Theorie der Polyeder: unter Einschlu{\ss} der Elemente der Topologie, volume 41, Springer-Verlag, 2013.