Bounds on the rate of disjunctive codes

• Published in: Problems of information transmission Vol. 50; no. 1; pp. 27 - 56
• Main Authors: D’yachkov, A. G., Vorob’ev, I. V., Polyansky, N. A., Shchukin, V. Yu
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 2014
• Subjects: Coding Theory; Communications Engineering; Control; Electrical Engineering; Engineering; Information Storage and Retrieval; Networks; Systems Theory; Asymptotic Equality; Information Transmission; Random Code; Asymptotic Inequality; Combinatorial Code Theory
• DOI: 10.1134/S0032946014010037
• EISSN: 1608-3253
• ISSN: 0032-9460

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 corresponding binary code is called a disjunctive
s
-code. This paper is aimed at improving previously known and obtaining new bounds on the rate of these codes. The most interesting of the new results is a lower bound on the rate of disjunctive (
s, ℓ
) cover-free codes obtained by random coding over the ensemble of binary constant-weight codes; its ratio to the best known upper bound converges as
s
→ ∞, with an arbitrary fixed
ℓ
≥ 1, to the limit 2
e
−2
= 0.271 ... In the classical case of
ℓ
= 1, this means that the upper bound on the rate of disjunctive
s
-codes constructed in 1982 by D’yachkov and Rykov is asymptotically attained up to a constant factor
a
, 2
e
−2
≤
a
≤ 1.