Abstract
We consider list decoding in the zero-rate regime for two cases - the binary alphabet and the spherical codes in Euclidean space. Specifically, we study the maximal τ ϵ [0,1] for which there exists an arrangement of M balls of relative Hamming radius τ in the binary hypercube (of arbitrary dimension) with the property that no point of the latter is covered by L or more of them. As M → ∞ the maximal τ decreases to a well-known critical value τ L . In this paper, we prove several results on the rate of this convergence. For the binary case, we show that the rate is Θ (M -1 ) when L is even, thus extending the classical results of Plotkin and Levenshtein for L=2. For L=3 , the rate is shown to be Θ (M -(2/3) ). For the similar question about spherical codes, we prove the rate is Ω (M -1 ) and O(M- (2L/L 2 -L+2) ).
Original language | English (US) |
---|---|
Article number | 8456617 |
Pages (from-to) | 1657-1667 |
Number of pages | 11 |
Journal | IEEE Transactions on Information Theory |
Volume | 65 |
Issue number | 3 |
DOIs | |
State | Published - Mar 2019 |
All Science Journal Classification (ASJC) codes
- Information Systems
- Computer Science Applications
- Library and Information Sciences
Keywords
- Euclidean space
- Hamming space
- List decoding
- error correction codes