Coded Categorization in Massive Random Access

Ryan Song, Kareem M. Attiah, Wei Yu

Abstract: This paper considers a massive random access scenario in which a small set of k users out of a large number of n potential users are active at any given time, and a central base-station wishes to send a common message to the active users in order to label them into a finite number of categories. Specifically, given c possible categories, the base-station wishes to send label ℓ to a set of k <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</inf> users, where ℓ ∈ {1, …, c} and $\sum\nolimits_{\ell = 1}^c {{k_\ell } = k} $. Assuming that n, k <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> , …, k <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</inf> are fixed, we ask: what is the minimum rate of the common message that the base-station needs to send so that the correct label is received at each of the k active users? This paper shows that instead of a conventional scheme of listing the indices of the users followed by their labels, which requires a common message rate of $k\left( {\log (n) + H\left( {\frac{{{k_1}}}{k}, \ldots ,\frac{{{k_c}}}{k}} \right)} \right)$ bits, it is possible to construct a fixed-length common message code with a rate of just $kH\left( {\frac{{{k_1}}}{k}, \ldots ,\frac{{{k_c}}}{k}} \right)$ bits plus a term that scales in n as O(log log(n)) for fixed k <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> , …, k <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</inf> , where H(•) is the entropy of a probability distribution. If a variable-length code is permitted, the minimum common message rate is characterized as $kH\left( {\frac{{{k_1}}}{k}, \ldots ,\frac{{{k_c}}}{k}} \right) + O(1)$ bits, with no dependence on n. Finally, if k <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> , …, k <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</inf> deviate from the values for which the common message is designed, an additional cost per user equal to a Kullback-Leibler divergence term would be incurred.