7 research outputs found
Sharper Upper Bounds for Unbalanced Uniquely Decodable Code Pairs
Two sets form a Uniquely Decodable Code Pair
(UDCP) if every pair , yields a distinct sum , where
the addition is over . We show that every UDCP , with and , satisfies . For sufficiently small , this bound significantly
improves previous bounds by Urbanke and Li~[Information Theory Workshop '98]
and Ordentlich and Shayevitz~[2014, arXiv:1412.8415], which upper bound
by and , respectively, as approaches .Comment: 11 pages; to appear at ISIT 201
An Upper Bound on the Sizes of Multiset-Union-Free Families
Let and be two families of subsets of an
-element set. We say that and are
multiset-union-free if for any and the multisets and are different, unless
both and . We derive a new upper bound on the maximal sizes of
multiset-union-free pairs, improving a result of Urbanke and Li.Comment: A shorter ISIT conference version titled "VC-Dimension Based Outer
Bound on the Zero-Error Capacity of the Binary Adder Channel" is availabl
A Note on the Probability of Rectangles for Correlated Binary Strings
Consider two sequences of independent and identically distributed fair
coin tosses, and , which are
-correlated for each , i.e. .
We study the question of how large (small) the probability can be among all sets of a given cardinality.
For sets it is well known that the largest (smallest)
probability is approximately attained by concentric (anti-concentric) Hamming
balls, and this can be proved via the hypercontractive inequality (reverse
hypercontractivity). Here we consider the case of . By
applying a recent extension of the hypercontractive inequality of
Polyanskiy-Samorodnitsky (J. Functional Analysis, 2019), we show that Hamming
balls of the same size approximately maximize in
the regime of . We also prove a similar tight lower bound, i.e.
show that for the pair of opposite Hamming balls approximately
minimizes the probability
Zero-error communication over adder MAC
Adder MAC is a simple noiseless multiple-access channel (MAC), where if users
send messages , then the receiver receives with addition over . Communication over the
noiseless adder MAC has been studied for more than fifty years. There are two
models of particular interest: uniquely decodable code tuples, and -codes.
In spite of the similarities between these two models, lower bounds and upper
bounds of the optimal sum rate of uniquely decodable code tuple asymptotically
match as number of users goes to infinity, while there is a gap of factor two
between lower bounds and upper bounds of the optimal rate of -codes.
The best currently known -codes for are constructed using
random coding. In this work, we study variants of the random coding method and
related problems, in hope of achieving -codes with better rate. Our
contribution include the following. (1) We prove that changing the underlying
distribution used in random coding cannot improve the rate. (2) We determine
the rate of a list-decoding version of -codes achieved by the random
coding method. (3) We study several related problems about R\'{e}nyi entropy.Comment: An updated version of author's master thesi