A Rate-Optimal Construction of Codes with Sequential Recovery with Low Block Length

An erasure code is said to be a code with sequential recovery with parameters <tex>$r$</tex> and t, if for any <tex>$s$</tex> ≤ <tex>$t$</tex> erased code symbols, there is an s-step recovery process in which at each step we recover exactly one erased code symbol by contacting at most <tex>$r$</tex> other code symbols. In this paper, we give a construction of binary codes with sequential recovery that are rate-optimal for any value of <tex>$t$</tex> and any value <tex>$r$</tex> ≥ 3. Our construction is based on construction of certain kind of tree-like graphs with girth <tex>$t$</tex> + 1. We construct these graphs and hence the codes recursively.

[1]  P. Vijay Kumar,et al.  Codes with locality for two erasures , 2014, 2014 IEEE International Symposium on Information Theory.

[2]  Moshe Morgenstern,et al.  Existence and Explicit Constructions of q + 1 Regular Ramanujan Graphs for Every Prime Power q , 1994, J. Comb. Theory, Ser. B.

[3]  Cheng Huang,et al.  On the Locality of Codeword Symbols , 2011, IEEE Transactions on Information Theory.

[4]  A. Lubotzky,et al.  Ramanujan graphs , 2017, Comb..

[5]  F. Lazebnik,et al.  A new series of dense graphs of high girth , 1995, math/9501231.

[6]  Xavier Dahan,et al.  Regular graphs of large girth and arbitrary degree , 2011, Comb..

[7]  Frédérique Oggier,et al.  Self-repairing homomorphic codes for distributed storage systems , 2010, 2011 Proceedings IEEE INFOCOM.

[8]  Sriram Vishwanath,et al.  Cooperative local repair in distributed storage , 2014, 2014 48th Annual Conference on Information Sciences and Systems (CISS).

[9]  Chau Yuen,et al.  Binary Locally Repairable Codes - Sequential Repair for Multiple Erasures , 2016, 2016 IEEE Global Communications Conference (GLOBECOM).

[10]  Dimitris S. Papailiopoulos,et al.  Locally Repairable Codes , 2014, IEEE Trans. Inf. Theory.

[11]  Balaji Srinivasan Babu,et al.  Binary codes with locality for multiple erasures having short block length , 2016, 2016 IEEE International Symposium on Information Theory (ISIT).

[12]  Balaji Srinivasan Babu,et al.  A tight rate bound and a matching construction for locally recoverable codes with sequential recovery from any number of multiple erasures , 2017, 2017 IEEE International Symposium on Information Theory (ISIT).

[13]  Chau Yuen,et al.  Locally Repairable Codes with Functional Repair and Multiple Erasure Tolerance , 2015, ArXiv.

[14]  G. Exoo,et al.  Dynamic Cage Survey , 2011 .

[15]  Giuliana P. Davidoff,et al.  Elementary number theory, group theory, and Ramanujan graphs , 2003 .

[16]  Wentu Song,et al.  On Sequential Locally Repairable Codes , 2018, IEEE Transactions on Information Theory.

[17]  Balaji Srinivasan Babu,et al.  Binary Codes with Locality for Four Erasures , 2016, ArXiv.