  We consider the two scenarios of communicating a pair $S_{1},S_{2}$ of
correlated sources over multiple access (MAC) and interference channels (IC)
respectively. We undertake a Shannon theoretic study and focus on
achievability, i.e., characterizing sufficient conditions. In the absence of a
Ga\'cs-K\"orner-Witsenhausen common part, the current known single-letter (S-L)
coding schemes are constrained to the S-L long Markov Chain (LMC)
$X_{1}-S_{1}-S_{2}-X_{2}$. Taking the lead of Dueck's example [Dueck, Mar
1981], we recognize that the latter constraint is debilitating, leading to
sub-optimality of S-L coding schemes. The goal of our work is to design a
coding scheme wherein (i) the choice of channel input at time $t$ is based on
multiple source symbols, and is yet ii) amenable to performance
characterization via S-L expressions. In this article, we present the first
part of our findings. We propose a new separation-based coding scheme based on
a fixed block-length (B-L) codes that enables choice of $X_{jt}$ - the symbol
input on the channel by encoder $j$ at time $t$ - to be based on a generic
number $l$ of source symbols $S_{j}^{l}$, thus permitting correlation of the
input symbols $X_{1},X_{2}$ through a multi-letter LMC
$X_{1}-S_{1}^{l}-S_{2}^{l}-X_{2}$. By carefully stitching together S-L coding
techniques we devise a multi-letter coding scheme. We characterize an inner
bound to its performance via a S-L expression and prove that the derived inner
bound is strictly larger than the current known largest inner bounds for both
the MAC and IC problems based on S-L coding schemes. In the second part of our
work, we propose to enlarge the inner bound derived in this article by
incorporating the technique of inducing source correlation onto channel inputs
[Cover, El Gamal and Salehi, Nov 1980].
