The Finite Field Multi-Way Relay Channel with Correlated Sources: Beyond Three Users

Consider $m$ independent and identically distributed (i.i.d.) drawings of a tuple of correlated discrete finite random variables $(W_{1},W_{2},\dotsc,W_{L})$, i.e., $\{(W_{1}[t],W_{2}[t],\dotsc,W_{L}[t])\}_{t=1}^{m}$. The message of user $i$ is given by $\boldsymbol{W}_{i}=(W_{i}[1],W_{i}[2],\dotsc,W_{i}[m])$.

Each channel use of the finite field MWRC consists of an uplink and $L$ downlinks, characterized by

where $X_{\ell},Y_{\ell},N_{\ell}$, for all $\ell\in\{0,1,\dotsc,L\}$, each take values in a finite field $\mathcal{F}$, $\oplus$ denotes addition over $\mathcal{F}$, $X_{\ell}$ is the channel input from node $\ell$, $Y_{\ell}$ is the channel output received by node $\ell$, and $N_{\ell}$ is the receiver noise at node $\ell$. The noise $N_{\ell}$ is arbitrarily distributed, but is i.i.d. for each channel use. We have used the subscript $i$ to denote a user and the subscript $\ell$ to denote a node (which can be a user or the relay).

Consider block codes for which the users exchange $m$ message tuples in $n$ channel uses. [Encoding:] The $t$-th transmitted channel symbol of node $\ell$ is a function of its message and the $(t-1)$ symbols it previously observed on the downlink: $X_{\ell}[t]=f_{\ell}[t](\boldsymbol{W}_{\ell},Y_{\ell}[1],Y_{\ell}[2],\dotsc,Y_{\ell}[t-1])$, for all $\ell\in\{0,1,\dotsc,L\}$ and for all $t\in\{1,2,\dotsc,n\}$. As the relay has no message, we set $\boldsymbol{W}_{0}=\varnothing$. [Decoding:] The messages decoded by user $i$ are a function of its message and the symbols it observed on the downlink: $(\hat{\boldsymbol{W}}_{j,i}:\forall j\in\{1,\dotsc,L\}\setminus\{i\})=h_{i}(\boldsymbol{W}_{i},Y_{i}[1],Y_{i}[2],\dotsc,Y_{i}[n])$, for all $i\in\{1,2,\dotsc,L\}$.

Let $P_{\text{e}}$ denote the probability that $\hat{\boldsymbol{W}}_{j,i}\neq\boldsymbol{W}_{i}$ for any $i\neq j$. The rate (or bandwidth expansion factor) of the code is the ratio of channel symbols to source symbols, $\kappa=n/m$. The rate $\kappa$ is said to be achievable if the following is true: for any $\epsilon>0$, there exists a block code, with $n$ and $m$ sufficiently large and $n/m=\kappa$, such that $P_{\text{e}}<\epsilon$.

For atoms with weight one,

Summing the contributions from all atoms with weight one to all $r_{i}\in\boldsymbol{r}^{\mathcal{S}}$,

We fix $K$. Consider the contributions from all atoms with weight $K$ to a particular $r_{i}$, $i\in\mathcal{S}$ (one column in the hashed region in Fig. 2). There are

• [$\mathsf{O1}$]

  $\binom{L-1}{K-1}$ contributions with coefficient $\frac{L-K}{L-1}$ [from atoms where $i\in\mathcal{K}$; there are $\binom{L-1}{K-1}$ ways to select the other $(K-1)$ elements in $\mathcal{K}$ from $\{1,2,\dotsc,L\}\setminus\{i\}$], and

• [$\mathsf{O2}$]

  $\binom{L-1}{L-K-1}$ contributions with coefficient $-\frac{K-1}{L-1}$ [from atoms where $i\in\mathcal{K}^{\text{c}}$; there are $\binom{L-1}{L-K-1}$ ways to select the other $(L-K-1)$ elements in $\mathcal{K}^{\text{c}}$ from $\{1,2,\dotsc,L\}\setminus\{i\}$].

There are $\binom{L}{K}$ atoms with weight $K$. We can check that $\binom{L}{K}=\binom{L-1}{K-1}+\binom{L-1}{L-K-1}$.

Since observations $\mathsf{O1}$ and $\mathsf{O2}$ are true for each $r_{i}\in\boldsymbol{r}^{\mathcal{S}}$, we have the following contributions from all atoms with weight $K$ to $\boldsymbol{r}^{\mathcal{S}}$ (the hashed region in Fig. 2): There are

• [$\mathsf{O3}$]

  $S\cdot\binom{L-1}{K-1}$ contributions with coefficient $\frac{L-K}{L-1}$, and

• [$\mathsf{O4}$]

  $S\cdot\binom{L-1}{L-K-1}$ contributions with coefficient $-\frac{K-1}{L-1}$.

For an atom $a(\mathcal{K})$, we say that the atom is active if $\mathcal{K}\subseteq\mathcal{S}$; otherwise, i.e., $\mathcal{K}\nsubseteq\mathcal{S}$, the atoms is said to be inactive,

Consider the contributions from a particular active atom $a(\mathcal{K})$ to $\boldsymbol{r}^{\mathcal{S}}$ (one row in the hashed region in Fig. 2). We have the following contributions from this atom to $\boldsymbol{r}^{\mathcal{S}}$:

• [$\mathsf{O5}$]

  Since $\mathcal{K}\subseteq\mathcal{S}$, $K$ cells in the hashed row each contribute $\frac{L-K}{L-1}\mu^{*}(a(\mathcal{K}))$.

• [$\mathsf{O6}$]

  The remaining $(S-K)$ cells each contribute $-\frac{K-1}{L-1}\mu^{*}(a(\mathcal{K}))$.

Summing the contributions from this active atom to $\boldsymbol{r}^{\mathcal{S}}$,

For any fixed $\mathcal{S}$, there are $\binom{S}{K}$ active atoms with weight $K$ (different ways of choosing $\mathcal{K}\subseteq\mathcal{S}$), and observations $\mathsf{O5}$ and $\mathsf{O6}$ are true for each active atom. Combining $\mathsf{O3}$–$\mathsf{O6}$, we can further categorize the contributions from all atoms with weight $K$ to $\boldsymbol{r}^{\mathcal{S}}$:

• [$\mathsf{O7}$]

  Out of the $S\cdot\binom{L-1}{K-1}$ contributions with coefficient $\frac{L-K}{L-1}$, $K\cdot\binom{S}{K}$ of them are from active atoms.

• [$\mathsf{O8}$]

  Out of the $S\cdot\binom{L-1}{L-K-1}$ contributions with coefficient $-\frac{K-1}{L-1}$, $(S-K)\binom{S}{K}$ of them are from active atoms.

Now, summing the contributions from all (active and inactive) atoms with weight $K$ to $\boldsymbol{r}^{\mathcal{S}}$, we have

where

where $\alpha(L,S,K)$ is defined in (9) and $\beta(L,S,K)$ in (10). If the ABCMI condition is satisfied, then $\eta\geq 0$.

Consider the contributions from all atoms with weight $K$ to $r_{i}$, for some $i$. From observations $\mathsf{O1}$ and $\mathsf{O2}$, we know that there are

• •

  $\binom{L-1}{K-1}$ contributions with coefficient $\frac{L-K}{L-1}$, and

• •

  $\binom{L-1}{L-K-1}$ contributions with coefficient $-\frac{K-1}{L-1}$.

Summing these contributions, we have for any $i$ that

Since $\alpha(L,S,K)\geq\beta(L,S,K)\geq 0$, the ABCMI condition implies that $\overline{\mu}_{K}\leq\left[1+\frac{1}{K-1}\right]\underline{\mu}_{K}=\frac{K}{K-1}\underline{\mu}_{K}$ for all $K\in\{2,3,\dotsc,L-1\}$. Hence, the inequality in (22) follows.

Substituting (19), (21b), and (22) into (17), if the sources have ABCMI, then

This completes the proof of Theorem 2. $\hfill\blacksquare$

We have the following source coding result for correlated sources:

The above result is obtained by combining the results for (i) source coding for correlated sources [7, Thm. 2], and (ii) the three-user noiseless MWRC [1, Sec. II.B.1]. Note that the relay does not participate in the source code, in contrast to the setup of [1]. Instead, the relay participates in the channel code.

We have the following channel coding result for the finite field MWRC [2]:

We propose the following separate source-channel coding scheme. Fix $mr_{i}=nR_{i}$. Let $(D_{1},\dotsc,D_{L})$, where each $D_{i}$ is i.i.d. and uniformly distributed on $\{1,\dotsc,2^{mr_{i}}\}$. We call $D_{i}$ a dither. The dithers are made known to all users. User $i$ performs source coding to compresses its message $\mathbf{W}_{i}$ to an index $M_{i}^{\prime}\in\{1,\dotsc,2^{mr_{i}}\}$ and computes $M_{i}=M_{i}^{\prime}+D_{i}\mod 2^{mr_{i}}$. The random variables $(M_{1},\ldots,M_{L})$ are independent, with $M_{i}$ being uniformly distributed on $\{1,\dotsc,2^{nR_{i}}\}$. The nodes then perform channel coding with $M_{i}$ as inputs. If (23) is satisfied, then each user $i$ can recover $\{M_{j}:\forall j\in\{1,\dotsc,L\}\setminus\{i\}\}$, from which it can obtain $\{M_{j}^{\prime}:\forall j\in\{1,\dotsc,L\}\setminus\{i\}\}$. If (4) is satisfied, then each user $i$ can recover all other users’ messages $\{\boldsymbol{W}_{j}:\forall j\in\{1,\dotsc,L\}\setminus\{i\}\}$, using the indices $M_{j}^{\prime}$’s and its own message $\boldsymbol{W}_{i}$. This means the rate $\kappa=n/m$ is achievable.

Using the above coding scheme, we have the following: