On the Capacity of the Finite Field Counterparts of Wireless Interference Networks

This paper explores how degrees of freedom (DoF) results from wireless networks can be translated into capacity or linear capacity results for their finite field counterparts that arise in network coding applications. The main insight is that scalar (SISO) finite field channels over
\(\mathbb {F}_{\!p^n}\)
are analogous to
\(n\times n\)
vector (MIMO) channels in the wireless setting, but with an important distinction—there is additional structure due to finite field arithmetic, which enforces commutativity of matrix multiplication and limits the channel diversity to
\(n\)
, making these channels similar to diagonal channels in the wireless setting. Within the limits imposed by the channel structure, the DoF optimal precoding solutions for wireless networks can be translated into capacity or linear capacity optimal solutions for their finite field counterparts. This is shown through the study of capacity of the 2-user X channel and linear capacity of the 3-user interference channel. Besides bringing the insights from wireless networks into network coding applications, the study of finite field networks over
\(\mathbb {F}_{\!p^n}\)
also touches upon important open problems in wireless networks (finite SNR, finite diversity scenarios) through interesting parallels between
\(p\)
and SNR, and
\(n\)
and diversity.