Extended Placement Delivery Arrays for Multi-Antenna Coded Caching Scheme
K. K. Krishnan Namboodiri, Elizabath Peter, B. Sundar Rajan
Abstract
This work addresses the multi-antenna coded caching problem where a server with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> transmit antennas communicates to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> users through a wireless broadcast link. In the problem setting, the server has a library of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> files, and each user is equipped with a dedicated cache of capacity <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> . A novel solution for the multi-antenna coded caching problem is obtained by designing a combinatorial structure called an extended placement delivery array (EPDA). It is shown that the placement delivery arrays known for the centralized coded caching scheme are a special class of EPDAs with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L=1$ </tex-math></inline-formula> . Furthermore, three constructions of EPDAs are proposed for the settings: a) <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K = t+L$ </tex-math></inline-formula> , b) <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K = nt+ (n-1)L;L\geq t, n\geq 2$ </tex-math></inline-formula> , and c) <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K,L,t$ </tex-math></inline-formula> such that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$t + L\leq K$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$t = KM/N$ </tex-math></inline-formula> is an integer. The multi-antenna schemes resulting from the first two constructions achieve the optimal degrees of freedom (DoF) <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$t+L$ </tex-math></inline-formula> with a subpacketization number -the number of subfiles into which a file is divided- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K/\text {gcd}(K,t,L)$ </tex-math></inline-formula> , which is lower than the subpacketization number of the existing schemes. The scheme obtained from the third construction also achieves the optimal DoF with a subpacketization number <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\binom {K/{\gamma }}{(t+L)/{\gamma }}\left ({{t+L}}\right)/{\gamma }$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\gamma =\text {gcd}(K,t,L)$ </tex-math></inline-formula> .