Sharper upper bounds for unbalanced Uniquely Decodable Code Pairs

Per Austrin, Petteri Kaski, Mikko Koivisto, Jesper Nederlof

Abstract: Two sets A, B ⊆ {0, 1} <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> form a Uniquely Decodable Code Pair (UDCP) if every pair a ∈ A, b ∈ B yields a distinct sum a+b, where the addition is over ℤ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> . We show that every UDCP A, B, with |A| = 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(1−ε)n</sup> and |B| = 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">βn</sup> , satisfies equation. For sufficiently small ε, 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 ε approaches 0.