An Upper Bound on the Sizes of Multiset-Union-Free Families

Abstract

Let F 1 and F 2 be two families of subsets of an n-element set. We say that F 1 and F 2 are multiset-union-free if for any A, B ∈ F 1 and C, D ∈ F 2 the multisets A C and B D are different, unless both A = B and C = D. We derive a new upper bound on the maximal sizes of multiset-union-free pairs, improving a result of Urbanke and Li.

Sharper Upper Bounds for Unbalanced Uniquely Decodable Code Pairs

“…We show that every UDCP A, B, with |A| = 2 (1−ǫ)n and |B| = 2 βn , satisfies β ≤ 0.4228 + √ ǫ. For sufficiently small ǫ, 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 β by 0.4921 and 0.4798, respectively, as ǫ approaches 0.…”

“…The intuition behind our main bound (and, partially, the bounds of Urbanke and Li [15] and Ordentlich and Shayevitz [13]) is as follows. The above strategy does not give a good bound if A and B are antipodal Hamming balls: the studied probability is very small in this case, so the upper bound is not really stringent.…”

“…On a high level, their approach works as follows: a result of van Tilborg [18] (see also Lemma 1 below) shows there are not many pairs (a, b) ∈ A × B of small Hamming distance, and if A and B are sufficiently large, then the number of such pairs is lower bounded by an isoperimetric inequality for which the authors use Harper's theorem. Later, this result was improved to β ≤ 0.4798 by Ordentlich and Shayevitz [13]. Their proof idea is somewhat more involved: the authors give a procedure that, given a UDCP A, B ⊆ {0, 1} n , constructs another UDCP C, C ∈ {0, 1} (1−γ)n with some γ > 0.…”

“…The existence of such a subset is proved using a variant of the Sauer-Perles-Shelah lemma. Unfortunately, both the referred bounds [13,15] converge fast to (1 − ǫ) + β ≤ 1.5 as ǫ increases (see Figure 1 of Ordentlich and Shayevitz [13]).…”

Sharper upper bounds for unbalanced Uniquely Decodable Code Pairs

“…On a high level, their approach works as follows: a result of van Tilborg [12] (see Lemma 1 below) shows there are not many pairs (a, b) ∈ A×B of small Hamming distance, and if A and B are sufficiently large, then the number of such pairs is bounded from below by an isoperimetric inequality for which the authors use Harper's theorem. Later, this result was improved to β ≤ 0.4798 by Ordentlich and Shayevitz [13]. Their proof idea is somewhat more involved: the authors give a procedure that, given a UDCP A, B ⊆ {0, 1} n , constructs another UDCP C, D ∈ {0, 1} (1−γ)n of comparable size for some γ > 0.…”