Outer Code Designs for Augmented and Local-Global Polar Code Architectures

In conventional polar code design, $N$ independent copies of a channel $W$ are combined in a recursive manner into a vector channel $W_{N}$, which is then split into $N$ channels ${W_{N}^{(i)},\;1\leq{i}\leq{N}}$, referred to as bit-channels. The Bhattacharyya parameter $Z(W_{N}^{(i)})$ is used to identify the quality of bit-channel $i$. A polar code of rate $R{=}\frac{K}{N}$ selects the $K$ most reliable bit-channels (with the smallest $Z(W_{N}^{(i)})$) to input information bits, and the remaining bit-channel inputs are frozen to zero. We use ${\mathcal{A}}$ to denote the set of information indices, and ${\mathcal{F}{=}\mathcal{A}^{c}}$ to denote the frozen indices. Let $G{=}F^{\bigotimes n}$ be the $N{\times}N$ matrix that is the $n$-th Kronecker power of $F{=}\left[\begin{array}[]{cc}1&0\\ 1&1\\ \end{array}\right]$, where $n{=}\log_{2}N$. The polar encoding process is specified by $x{=}uG$, where $x,u\in\mathbb{F}^{N},G\in\mathbb{F}^{N\times N}$.

Arıkan showed that a systematic encoder can be realized that maps information bits to positions in the set ${\mathcal{B}}{=}{\mathcal{A}}$ in the codeword $x$. To be specific, $u_{\mathcal{A}^{c}}$ is set to 0, $x_{\mathcal{B}}$ is set to the information vector, and $u_{\mathcal{A}}$ and $x_{\mathcal{B}^{c}}$ are found by solving a system of equations [2].

Our focus in this paper is on concatenated code architectures in which all component codes are polar codes. The augmented and flexible length architectures were introduced in [6]. In an augmented polar code, a short, rate $R_{0}=\frac{K_{0}}{N_{0}}$ auxiliary outer polar code $G_{0}$ is connected to an inner polar code $G_{1}$ of length $N_{1}$. The $N_{0}$ bits of the outer codeword are assigned to the semipolarized channels within the inner code (through an interleaver). An additional $K_{1}$ message bits are assigned to the good bit-channels within the inner code. The total code rate for the augmented structure is $R_{aug}=\frac{K_{0}+K_{1}}{N_{1}}$.

In the flexible length architecture, two inner codes $G_{1},G_{2}$ of length $N_{1},N_{2}$ are coupled through a rate $R_{0}=\frac{K_{0}}{N_{0}}$ auxiliary outer code $G_{0}$. Information words of length $K_{1},K_{2}$ are assigned to the good bit-channels of the two inner codes, respectively. The outer codeword is divided into two parts which are assigned to the semipolarized bit-channels of the inner codes. The total encoding rate for the flexible length structure is $R_{flex}=\frac{K_{0}+K_{1}+K_{2}}{N_{1}+N_{2}}$.

Inspired by the flexible length architecture, the local-global polar code architecture, introduced in [7], connects multiple inner codes $G_{1},...,G_{M}$ through a systematic outer polar code. We asssume these codes have the same length $N_{i}=N,i=1,\ldots,M$. A word of $K_{b}$ information bits is divided into $M$ parts of $K_{b_{1}},\ldots,K_{b_{M}}$ bits that are assigned to good bit-channels within the inner codes. The $K_{a}$ outer information bits are divided into $M$ parts of $K_{a_{1}},\ldots,K_{a_{M}}$ bits that are mapped to semipolarized bit-channels in the $M$ inner codes, respectively. The $P_{a}$ parity bits of the outer codeword are similar partitioned into $M$ parts of $P_{a_{1}},\ldots,P_{a_{M}}$ bits and mapped to remaining semipolarized bit-channels within the inner codes. This architecture supports local decoding of information bits $K_{a_{i}},K_{b_{i}}$ within each inner code $G_{i}$, with the option of improved decoding of the $M$ inner codewords via global decoding using the outer code.

We briefly review the stopping set analysis of polar codes in [3]. Recall that a stopping set (SS) is a non-empty set of variable nodes such that each neighboring check node is connected to this set at least twice. A stopping tree (ST) is a SS that contains one and only one information bit, i.e., variable node in the leftmost stage of the sparse polar code factor graph. For each information bit $i$, there is a unique stopping tree denoted by $ST(i)$. The size of the leaf set (variable nodes on the rightmost stage) of $ST(i)$ is denoted by $f(i)$. Only variable nodes on the right are observed nodes, with all other variables nodes hidden. The set of observed variable nodes in a SS form a variable-node SS (VSS). Accordingly, we define a minimum VSS (MVSS) to be a VSS with a minimum number of observed variable nodes, among all the VSSs. The size of a MVSS is the stopping distance of the code. For any given set $\mathcal{J}\subseteq\mathcal{A}$, we denote the set of SSs whose information nodes are precisely $\mathcal{J}$ as $SS(\mathcal{J})$. The set of observed variable nodes in each of these SSs is a VSS for $\mathcal{J}$, and the collection of these VSSs is denoted as $VSS(\mathcal{J})$. A minimum size VSS in $VSS(\mathcal{J})$ is called a minimum VSS for $\mathcal{J}$, denoted $MVSS(\mathcal{J})$.

Consider the augmented polar code structure in Fig. 1, where an inner polar code is concatenated with an outer polar code. Let $v_{0}(i)$ denote the $i$-th node on the leftmost stage within the original outer factor graph, and $v_{1}(j)$ be the $j$-th node on the leftmost stage within the original inner factor graph. Let $\mathcal{J}_{i}$ denote the set of inner information nodes such that each element in $\mathcal{J}_{i}$ is connected to one of the leaves in the stopping tree $ST(i)$ defined on the original outer factor graph. For example in Fig. 1, $\mathcal{J}_{3}$ is the set of nodes $\{{v_{1}(2),v_{1}(6)}\}$. Let $MVSS(\mathcal{J}_{i})$ be the MVSS defined by set $\mathcal{J}_{i}$ on the original inner factor graph. Let $G_{\mathcal{J}_{i}}$ denote the submatrix of the inner encoding matrix $G=F^{\bigotimes n_{1}}$ consisting of the rows that correspond to $\mathcal{J}_{i}$. Again taking Fig. 1 as an example, $MVSS(\mathcal{J}_{3})$ is the set of nodes $\{{x_{5},x_{6}}\}$. The resulting $G_{\mathcal{J}_{3}}$ consists of the second and sixth rows of inner encoding matrix $G=F^{\bigotimes 3}$. Note that $\{x_{1},x_{2},x_{5},x_{6}\}$ is also a VSS for $\mathcal{J}_{i}$, but it is not the minimum VSS. The following theorem gives a bound on the size of $MVSS(\mathcal{J}_{i})$.

In words, the function $g(\cdot)$ counts the number of columns in a matrix that have weight one. Thus, for any given information position $i$ we easily calculate the lower bound on the size of the MVSS that only contains $v_{0}(i)$ on the leftmost stage by looking at the generator matrix of the inner code.

Theorem 2 suggests a practical way to design outer polar codes based on the size of stopping sets. We first initialize an unfrozen set $\mathcal{O}$ for the outer code using the conventional DE algorithm, for example. Then we swap a specified number of unfrozen bits $i\in{\mathcal{O}}$ with the smallest “stopping distance” $g(G_{\mathcal{J}_{i}})$ with some positions $j\in\mathcal{O}^{c}$ such that $g(G_{\mathcal{J}_{j}})>g(G_{\mathcal{J}_{i}})$.

Before presenting the design algorithm, we introduce some notation. Let $Q$ be a length $N_{0}$ vector that contains the indices of bit channels ordered according to channel reliability calculated by DE. The indices are ordered by descending channel reliability, i.e., $Q(1)$ stores the index for the strongest bit channel, $Q(2)$ stores the index for the second strongest, and so on. Let $s$ denote the number of bits we are going to swap. For convenience, we denote $g(G_{\mathcal{J}_{i}})$ by $g(i)$. Let $K_{0}$ denote the size of the desired unfrozen set. Let $min_{s}(\cdot)$ be the function that returns the $s$-th smallest value in a vector, while $min(\cdot)$ returns the smallest value along with its index. Note that $s$ should be chosen such that there are more than $s$ frozen bits that have $g(\cdot)$ value larger than $min_{s}(g(Q(1)),...,g(Q(K_{0})))$. The detailed swapping algorithm is presented in Algorithm 1.

We can easily extend Theorem 2 to the case when $M$ inner codes are connected by a single outer code. For example, assume $M=2$ and $\mathcal{J}_{i}=\{\mathcal{J}_{i}^{1},\mathcal{J}_{i}^{2}\}$, where $\mathcal{J}_{i}^{1}$ and $\mathcal{J}_{i}^{2}$ are connected nodes in the first and second inner codes, respectively. Then $|MVSS(\mathcal{J}_{i})|\geq(g(G_{\mathcal{J}_{i}^{1}})+g(G_{\mathcal{J}_{i}^{2}}))$.

The design method of Algorithm 1 can be extended to the local-global polar code architecture, but some care is needed. The systematic outer polar code assigns $M$ information vectors $K_{a_{i}}$, $i{=}1,\ldots,M$ to the $M$ inner codes. Directly applying Algorithm 1 can potentially swap bits $K_{a_{i}}$ with $P_{a_{j}}$ ($i\neq j$), causing $[K_{a_{i}},K_{a_{j}}]$ to be assigned to the same inner code. For example, assume the unfrozen set $\mathcal{O}=\{3,4,7,8\}$ represents the most reliable positions according to DE, and $\mathcal{O}^{c}=\{1,2,5,6\}$. Then, if the partition of $\mathcal{O}$ is according to bit index, the first half of $\mathcal{O}$ will correspond to $K_{a_{1}}=\{3,4\}$ and the second half to $K_{a_{2}}=\{7,8\}$. If the parity bits are partitioned similarly, we have $P_{a_{1}}=\{1,2\}$ and $P_{a_{2}}=\{5,6\}$. If, after calculating the $G_{\mathcal{J}_{i}}$ value for each position, we swap positions 3 and 5, this would yield $K_{a_{1}}=\{4,5\}$. This assignment is now inconsistent with the local-global architecture because part of $K_{a_{1}}$ (position 5) is connected with the second inner code. To avoid this problem, one needs to carefully choose the connections in the local-global encoder to ensure that positions in $K_{a_{i}}$ are only swapped with positions in $P_{a_{i}}$.