# Bounds on the rate of disjunctive codes

@article{Dyachkov2013BoundsOT, title={Bounds on the rate of disjunctive codes}, author={Arkadii G. D'yachkov and Ilya Vorobyev and N. A. Polyansky and Vladislav Shchukin}, journal={Problems of Information Transmission}, year={2013}, volume={50}, pages={27-56} }

A binary code is said to be a disjunctive (s, ℓ) cover-free code if it is an incidence matrix of a family of sets where the intersection of any ℓ sets is not covered by the union of any other s sets of this family. A binary code is said to be a list-decoding disjunctive of strength s with list size L if it is an incidence matrix of a family of sets where the union of any s sets can cover no more that L − 1 other sets of this family. For L = ℓ = 1, both definitions coincide, and the…

## 48 Citations

### Almost Disjunctive List-Decoding Codes (two talks)

- Computer Science
- 2014

A random coding method based on the ensemble of binary constant-weight codes to obtain lower bounds on the capacity and error probability exponent of such codes.

### Almost disjunctive list-decoding codes

- Computer ScienceProbl. Inf. Transm.
- 2015

The random coding method on the ensemble of binary constant-weight codes is established and lower bounds on the capacity and error exponent of almost disjunctive sL-LD codes are established.

### Almost disjunctive list-decoding codes

- Computer ScienceProblems of Information Transmission
- 2015

The random coding method on the ensemble of binary constant-weight codes is established and lower bounds on the capacity and error exponent of almost disjunctive sL-LD codes are established.

### Almost cover-free codes

- Computer ScienceProbl. Inf. Transm.
- 2016

The most interesting result is the proof of a lower and an upper bound for the capacity of (s, l) ACF codes; the ratio of these bounds tends as s→∞ to the limit value log2e/(le).

### Bounds on the rate of separating codes

- Computer ScienceProblems of Information Transmission
- 2017

Bounds on the rate of binary (s, l) separating codes, the most important for applications, are studied in more detail and tables of numerical values of the best presently known bounds on the rates are given.

### Bounds on the rate of separating codes

- Computer Science
- 2017

Bounds on the rate of binary (s, l) separating codes, the most important for applications, are studied in more detail and tables of numerical values of the best presently known bounds on the rates are given.

### Almost disjunct matrices from codes and designs

- Computer Science, MathematicsArXiv
- 2015

A new connection is established between $(t,\epsilon)$-disjunct matrices and error correcting codes based on the dual distance of the codes and estimates of the parameters of codes that give rise to such schemes are derived.

### Low-weight superimposed codes and related combinatorial structures: Bounds and applications

- Computer ScienceTheor. Comput. Sci.
- 2020

### Symmetric disjunctive list-decoding codes

- Computer Science2015 IEEE International Symposium on Information Theory (ISIT)
- 2015

The main purpose of this work is to obtain bounds on the rate of these codes, which are a class of binary codes based on a symmetric disjunctive sum of binary symbols.

### Separable Codes for the Symmetric Multiple-Access Channel

- Computer Science2018 IEEE International Symposium on Information Theory (ISIT)
- 2018

This paper generalizes the problem and discusses upper and lower bounds on the rate of q-ary s-separable codes for models of noiseless symmetric MAC, i.e., at each time instant the output signal of MAC is a symmetric function of its input signals.

## 31 References

### New Results in the Theory of Superimposed Codes: Part I

- Mathematics, Computer Science
- 2000

The concept of a binary superimposed (s,)-code identified by a family of finite sets in which no intersection of sets is covered by the union of s others is introduced and discussed.

### Upper Bounds on the Rate of Superimposed ( s ; ` )-Codes Based on Engel ' s Inequality 1

- Computer Science
- 2006

This work improves upper bounds on the rate of superimposed (s; `) codes obtained in [3, 4], using an important combinatorial result of K. Engel to denote de nitional equalities.

### Asymptotic Upper Bound for the Rate of (w, r) Cover-Free Codes

- Computer Science, MathematicsProbl. Inf. Transm.
- 2003

A new recurrent inequality is obtained for the rate of (w, r) cover-free codes, which improves previously known upper bounds on the rate.

### Families of Finite Sets in which No Intersection of Sets Is Covered by the Union of s Others

- Computer ScienceJ. Comb. Theory, Ser. A
- 2002

This paper considers a generalization of the superimposed code concept called a binary superimposed ( s,l )-code which is identified by the incidence matrix of a family defined in the title, and discusses the constructions based on MDS-codes.

### Nonrandom binary superimposed codes

- Computer ScienceIEEE Trans. Inf. Theory
- 1964

In this paper some basic properties of nonrandom codes of this family are presented, and formulas and bounds relating the principal code parameters are derived.

### On optimal superimposed codes

- Computer Science, Mathematics
- 2004

This paper develops a method of constructing superimposed codes and proves that some superimposed code constructed in this way are optimal and can be used in cryptography as a concept of key distribution patterns.

### Information Theory, Combinatorics, and Search Theory

- Computer ScienceLecture Notes in Computer Science
- 2013

It is proven that the q-ary identification entropy HI,q(P ) is a lower bound for the average number L(P, P ) of expected checkings during the identification process, and an alteration of their scheme is discovered which strengthens this upper bound significantly.

### Interval Packing and Covering in the Boolean Lattice

- MathematicsCombinatorics, Probability and Computing
- 1996

Let be the hypergraph whose points are the subsets X of [n] := {1,…,n} with l≤ |X| ≤ u, l < u, and whose edges are intervals in the Boolean lattice of the form I = {C ⊆[n] : X⊆C⊆Y} where |X| = l, |Y|…

### New constructions of superimposed codes

- Computer ScienceIEEE Trans. Inf. Theory
- 2000

Applying a concatenation of the binary constant-weight error-correcting codes and the shortened RS codes, new constructions of superimposed codes are obtained.

### On the construction of (w, r) cover-free codes

- Computer Science, MathematicsProbl. Inf. Transm.
- 2009

It is proved that in this class there exists a sequence of (w, r) cover-free codes which has a nonzero limit rate for w, r = const.