Invertible Bloom Lookup Tables with Listing Guarantees

The Invertible Bloom Lookup Table (IBLT) is a probabilistic concise data structure for set representation that supports a listing operation as the recovery of the elements in the represented set. Its applications can be found in network synchronization and traffic monitoring as well as in error-correction codes. IBLT can list its elements with probability affected by the size of the allocated memory and the size of the represented set, such that it can fail with small probability even for relatively small sets. While previous works only studied the failure probability of IBLT, this work initiates the worst case analysis of IBLT that guarantees successful listing for all sets of a certain size. The worst case study is important since the failure of IBLT imposes high overhead. We describe a novel approach that guarantees successful listing when the set satisfies a tunable upper bound on its size. To allow that, we develop multiple constructions that are based on various coding techniques such as stopping sets and the stopping redundancy of error-correcting codes, Steiner systems, and covering arrays as well as new methodologies we develop. We analyze the sizes of IBLTs with listing guarantees obtained by the various methods as well as their mapping memory consumption. Lastly, we study lower bounds on the achievable sizes of IBLT with listing guarantees and verify the results in the paper by simulations

Editor’s note

This Special Issue, entitled Algebraic Combinatorics and Applications, of the Journal of Algebra, Combinatorics, Discrete Structures, and Applications, contains selected papers submitted by conference participants at the "Algebraic Combinatorics and Applications: The First Annual Kliakhandler Conference", Houghton, Michigan, USA, August 26 - 30, 2015, as well as two additional papers submitted in response to our call for papers. The conference took place on the campus of Michigan Technological University, and was attended by 43 researchers and graduate and postdoctoral students from USA, Canada, Croatia, Japan, South Africa, and Turkey. Funding for the conference was provided by a generous gift of Igor Kliakhandler, and a grant from the National Science Foundation. The conference brought together researchers and students interested in combinatorics and its applications, to learn about the latest developments, and explore different visions for future work and collaborations. Over thirty talks were presented on various topics from combinatorial designs, graphs, finite geometry, and their applications to error-correcting codes, network coding, information security, quantum computing, DNA codes, mobile communications, and tournament scheduling. The current Special Issue contains papers on covering arrays and their applications, group divisible designs, automorphism groups of combinatorial designs, covering number of permutation groups, tournaments, large sets of geometric designs, partitions, quasi-symmetric functions, resolvable Steiner systems, and weak isometries of Hamming spaces

Binary and Ternary Quasi-perfect Codes with Small Dimensions

The aim of this work is a systematic investigation of the possible parameters of quasi-perfect (QP) binary and ternary linear codes of small dimensions and preparing a complete classification of all such codes. First we give a list of infinite families of QP codes which includes all binary, ternary and quaternary codes known to is. We continue further with a list of sporadic examples of binary and ternary QP codes. Later we present the results of our investigation where binary QP codes of dimensions up to 14 and ternary QP codes of dimensions up to 13 are classified.Comment: 4 page