A set of quasi-uniform random variables X 1 , .
We address the question of constructing explicitly quasi-uniform codes from groups. We determine the size of the codebook, the alphabet and the minimum distance as a function of the corresponding group, both for abelian and some nonabelian groups. Potentials applications comprise the design of almost affine codes and non-linear network codes.
Abstract-We study the application of polar codes in deletion channels by analyzing the cascade of a binary erasure channel (BEC) and a deletion channel. We show how polar codes can be used effectively on a BEC with a single deletion, and propose a list decoding algorithm with a cyclic redundancy check for this case. The decoding complexity is O (N 2 log N), where N is the blocklength of the code. An important contribution is an optimization of the amount of redundancy added to minimize the overall error probability. Our theoretical results are corroborated by numerical simulations, which show that the list size can be reduced to one and the original message can be recovered with high probability as the length of the code grows.
Linear rank inequalities in 4 subspaces are characterized by Shannon-type inequalities and the Ingleton inequality in 4 random variables. Examples of random variables violating these inequalities have been found using finite groups, and are of interest for their applications in nonlinear network coding [1]. In particular, it is known that the symmetric group S5 provides the first instance of a group, which gives rise to random variables that violate the Ingleton inequality. In the present paper, we use group theoretic methods to construct random variables which violate linear rank inequalities in 5 random variables. In this case, linear rank inequalities are fully characterized [8] using Shannon-type inequalities together with 4 Ingleton inequalities and 24 additional new inequalities. We show that finite groups which do not produce violators of the Ingleton inequality in 4 random variables will also not violate the Ingleton inequalities for 5 random variables. We then focus on 2 of the 24 additional inequalities in 5 random variables and formulate conditions for finite groups which help us eliminate those groups that obey the 2 inequalities. In particular, we show that groups of order pq, where p, q are prime, always satisfy them, and exhibit the first violator, which is the symmetric group S4.
We propose a new model of asynchronous batch codes that allow for parallel recovery of information symbols from a coded database in an asynchronous manner, i.e. when different queries take different time to process. Then, we show that the graph-based batch codes studied by Rawat et al. are asynchronous. Further, we demonstrate that hypergraphs of Berge girth at least 4, respectively at least 3, yield graphbased asynchronous batch codes, respectively private information retrieval (PIR) codes. We prove the hypergraph-theoretic proposition that the maximum number of hyperedges in a hypergraph of a fixed Berge girth equals the quantity in a certain generalization of the hypergraph-theoretic (6,3)-problem, first posed by Brown, Erdős and Sós. We then apply the constructions and bounds by Erdős, Frankl and Rödl about this generalization of the (6,3)problem, known as the (3r-3,r)-problem, to obtain batch code constructions and bounds on the redundancy of the graph-based asynchronous batch and PIR codes. Finally, we show that the optimal redundancy ρ(k) of graph-based asynchronous batch codes of dimension k with the query size t = 3 is 2 √ k. Moreover, for a general fixed value of t ≥ 4, ρ(k) = O k 1/(2−ǫ) for any small ǫ > 0. For a general value of t ≥ 4, lim k→∞ ρ(k)/ √ k = ∞. Index Terms-primitive linear multiset batch codes, private information retrieval codes, extremal hypergraph theory, Turán theory, packing designs.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.