Math @ Duke

Publications [#335323] of Robert Calderbank
Papers Published
 Kadhe, S; Calderbank, R, Rate optimal binary linear locally repairable codes with small availability,
Ieee International Symposium on Information Theory Proceedings
(August, 2017),
pp. 166170, ISBN 9781509040964 [doi]
(last updated on 2018/11/13)
Abstract: © 2017 IEEE. A locally repairable code with availability has the property that every code symbol can be recovered from multiple, disjoint subsets of other symbols of small size. In particular, a code symbol is said to have (r, t)availability if it can be recovered from t disjoint subsets, each of size at most r. A code with availability is said to be rate optimal, if its rate is maximum among the class of codes with given locality, availability, and alphabet size. This paper focuses on rateoptimal binary, linear codes with small availability, and makes three contributions. First, it establishes tight upper bounds on the rate of binary linear codes with (r, 2) and (2, 3) availability. Second, it establishes a uniqueness result for binary rateoptimal codes, showing that for certain classes of binary linear codes with (r, 2) and (2, 3)availability, any rateoptimal code must be a direct sum of shorter rateoptimal codes. Finally, it presents a class of locally repairable codes associated with convex polyhedra, especially, focusing on the codes associated with the Platonic solids. It demonstrates that these codes are locally repairable with t = 2, and that the codes associated with (geometric) dual polyhedra are (coding theoretic) duals of each other.


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