Improving Uniquely Decodable Codes in Binary Adder Channels. arXiv:2312.11723
Preprint, arXiv:2312.11723 [math.CO] (2023).
Summary: We present a general method to modify existing uniquely decodable codes in the \(T\)-user binary adder channel. If at least one of the original constituent codes does not have average weight exactly half of the dimension, then our method produces a new set of constituent codes in a higher dimension, with a strictly higher rate. Using our method we improve the highest known rate for the \(T\)-user binary adder channel for all \(T \geq 2\). This information theory problem is equivalent to co-Sidon problems initiated by Lindström in the 1960s, and also the multi-set union-free problem. Our results improve the known lower bounds in these settings as well.
MSC:
| 05D40 | Probabilistic methods in extremal combinatorics, including polynomial methods (combinatorial Nullstellensatz, etc.) |
| 05C65 | Hypergraphs |
| 05D05 | Extremal set theory |
| 94A40 | Channel models (including quantum) in information and communication theory |
| 05B10 | Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.) |
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