Constructions of General Covering Designs

Given five positive integers $v, m,k,\lambda$ and $t$ where $v \geq k \geq t$ and $v \geq m \geq t,$ a $t$-$(v,k,m,\lambda)$ general covering design is a pair $(X,\mathcal{B})$ where $X$ is a set of $v$ elements (called points) and $\mathcal{B}$ a multiset of $k$-subsets of $X$ (called blocks) such that every $m$-subset of $X$ intersects (is covered by) at least $\lambda$ members of $\mathcal{B}$ in at least $t$ points. In this article we present new constructions for general covering designs and we generalize some others. By means of these constructions we will be able to obtain some new upper bounds on the minimum size of such designs.Comment: Section 3.2 revised and extended; plus some re-editing throughou

Low-Power Cooling Codes with Efficient Encoding and Decoding

A class of low-power cooling (LPC) codes, to control simultaneously both the peak temperature and the average power consumption of interconnects, was introduced recently. An $(n,t,w)$-LPC code is a coding scheme over $n$ wires that (A) avoids state transitions on the $t$ hottest wires (cooling), and (B) limits the number of transitions to $w$ in each transmission (low-power). A few constructions for large LPC codes that have efficient encoding and decoding schemes, are given. In particular, when $w$ is fixed, we construct LPC codes of size $(n/w)^{w-1}$ and show that these LPC codes can be modified to correct errors efficiently. We further present a construction for large LPC codes based on a mapping from cooling codes to LPC codes. The efficiency of the encoding/decoding for the constructed LPC codes depends on the efficiency of the decoding/encoding for the related cooling codes and the ones for the mapping

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

Métaheuristiques appliquées au problème de covering design

Résumé Un (v; k; t)-covering design est un ensemble de blocs (sous ensemble à k éléments d'un ensemble de référence V à v éléments) tel que tout sous-ensemble à t éléments de V soit contenu dans un des blocs. Considérant v, k et t, le problème de Covering Design consiste à trouver un covering contenant le moins de blocs possible. Pour résoudre le problème, nous avons adapté des métaheuristiques au Covering Design. Nous avons en particulier conçu un algorithme tabou avec diversication et un algorithme mémétique. Afin de rendre ces algorithmes plus rapides, nous les avons munis de nouvelles structures de données totalement incrémentales. Nos algorithmes de bas niveau sont devenus ainsi plus rapides et moins gourmands en espace mémoire. Nos algorithmes ont donc été capables de traiter des jeux de données que les précédentes métaheuristiques développées ne pouvaient pas tester. En matière de vitesse, nos algorithmes sont entre 10 et 100 fois plus rapides et l'accélération est d'autant plus élevé que les jeux de données à traiter sont gros. Nous avons testé nos algorithmes sur plus de 700 jeux de données. Nous avons ainsi réussi à trouver un meilleur covering design pour 77 jeux de données, dont 71 n'ont pas été améliorés par la suite.----------Abstract A (v; k; t)-covering design is a collection of k-subsets (called blocks) of a v-set V such that every t-subset of V is contained in at least one block. Given v, k and t, the goal of the Covering Design problem is to nd a covering made of a minimum number of blocks. In this paper, we present a new tabu algorithm with a mechanism of diversication and a new memetic algorithm for the solution of the problem. Our algorithms use a new implementation totally incremental designed in order to evaluate efficiently the performance of the neighbors of the current conguration. The new implementation is much less space-consuming than the currently used technique, making it possible to tackle much larger problem instances. It is also signicantly faster (between 10 and 100 times faster) and the speeding rate gets higher and higher as the size of the instances raises. We mesured the performance of our tabu algorithm trying more than 700 problem instances. Thanks to the improved data structures, our tabu algorithm was able to improve the upper bound of 77 problem instances and still hold the record for 71 of them

Covering Radius 1985-1994

We survey important developments in the theory of covering radius during the period 1985-1994. We present lower bounds, constructions and upper bounds, the linear and nonlinear cases, density and asymptotic results, normality, specific classes of codes, covering radius and dual distance, tables, and open problems