On optimal superimposed codes

Abstract

A (w, r ) cover-free family is a family of subsets of a finite set such that no intersection of w members of the family is covered by a union of r others. A binary (w, r ) superimposed code is the incidence matrix of such a family. Such a family also arises in cryptography as a concept of key distribution patterns. In this paper, we develop a method of constructing superimposed codes and prove that some superimposed codes constructed in this way are optimal.

Some new results on superimposed codes

“…We construct superimposed codes from super-simple designs which give us results better than superimposed codes constructed by other known methods. Next we prove the uniqueness of some optimal superimposed codes constructed in [4], [8], [10], and [15]. We finally improve numerical values of upper bounds for the asymptotic rate of some ðw; rÞ superimposed codes.…”

“…In this section, we prove that some optimal superimposed codes constructed in [4], [8], [10], and [15] are unique. For this purpose, we need information on the values of NðT; 1; 1Þ, NðT; 1; 2Þ, NðT; 1; 3Þ, and NðT; 2; 2Þ for small values of T. It is well known (the Sperner theorem) that NðT; 1; 1Þ is the smallest N such that…”

“…When the trivial superimposed code is not optimal, the computation of the value NðT; w; rÞ is a serious research problem. We list known values of NðT; w; rÞ which are less than the length of the corresponding trivial superimposed codes (for a proof, we refer to [10]): Nð6; 2; 2Þ ¼ Nð7; 2; 2Þ ¼ Nð8; 2; 2Þ ¼ 14; Nð9; 2; 2Þ¼ 18; Nð10; 2; 3Þ ¼ 30; and Nð11; 3; 3Þ ¼ 66: For larger values of T (resp. N), it seems very difficult to find exact values of NðT; w; rÞ (resp.…”

An Algorithm for Improving Throughput Guarantee of Topology-Transparent MAC Scheduling Strategy

“…WSN block [10,11] and superimposed code [12][13][14]. The aim of them is same, that is, to give birth to a cover-free set as MAC scheduling codes, which is the key to QoS guarantee.…”

Super‐simple resolvable balanced incomplete block designs with block size 4 and index 3

“…The existence of super-simple designs is an interesting extremal problem by itself, but there are also some useful applications. Such designs are used in construction of coverings [6], in construction of new designs [5], and in the construction of superimposed codes [19].…”