  Consider a receiver in a multi-user network that wishes to decode several
messages. Simultaneous joint typicality decoding is one of the most powerful
techniques for determining the fundamental limits at which reliable decoding is
possible. This technique has historically been used in conjunction with random
i.i.d. codebooks to establish achievable rate regions for networks. Recently,
it has been shown that, in certain scenarios, nested linear codebooks in
conjunction with "single-user" or sequential decoding can yield better
achievable rates. For instance, the compute-forward problem examines the
scenario of recovering $L \le K$ linear combinations of transmitted codewords
over a $K$-user multiple-access channel (MAC), and it is well established that
linear codebooks can yield higher rates. Here, we develop bounds for
simultaneous joint typicality decoding used in conjunction with nested linear
codebooks, and apply them to obtain a larger achievable region for
compute-forward over a $K$-user discrete memoryless MAC. The key technical
challenge is that competing codeword tuples that are linearly dependent on the
true codeword tuple introduce statistical dependencies, which requires careful
partitioning of the associated error events.
