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Recursive descriptions of polar codes

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  • Polar codes are recursive general concatenated codes. This property motivates a recursive formalization of the known decoding algorithms: Successive Cancellation, Successive Cancellation with Lists and Belief Propagation. Using such description allows an easy development of these algorithms for arbitrary polarizing kernels. Hardware architectures for these decoding algorithms are also described in a recursive way, both for Arıkan's standard polar codes and for arbitrary polarizing kernels.

    Mathematics Subject Classification: Primary: 94B05, 94B60; Secondary: 94B35.


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  • Figure 1.  A GCC representation of a polar code of length $\ell^n$ symbols constructed by a homogenous kernel according to Definition 1

    Figure 2.  Example 1's GCC representation (Arıkan's construction)

    Figure 3.  Representation of a polar code with kernel of $\ell = 2$ dimensions as a layered factor graph

    Figure 4.  Representation of a polar code with kernel of $\ell = 2$ dimensions as a layered factor graph (detailed version of Figure 3-recursion unfolded)

    Figure 5.  Normal factor graph representation of the $g(\cdot)$ block from Figures 3 and 4 for Arikan's $(u+v, v)$ construction

    Figure 6.  A GCC representation of the length $N=4^n$ bits mixed-kernels polar code $g^{(n)}(\cdot)$ described in Example 3

    Figure 7.  Decoding tree for $(u+v, v)$ polar code illustrating the decision space of the SC and SCL algorithms

    Figure 8.  Representation of SC as a sequential walk on a decoding tree

    Figure 9.  SCL ($L=4$) algorithm example of $(u+v, v)$ with $N=8$ bits (see Figure 7) illustrated on the right a decoding tree on the outer-codes of the structure ($\mathcal{C}_{0}, \mathcal{C}_{1}$). The left decoding tree expands each edge of the right tree into decoding-paths on the outer-codes of $\mathcal{C}_{0}$ and $\mathcal{C}_{1}$. The labels of the edges are the values of the outer-codes.

    Figure 10.  Normal factor graphs representations of polar codes kernels

    Figure 11.  Messages of BP algorithm

    Figure 12.  Normal factor graph representation for the first kernel of Example 4. This kernel is constructed by gluing inputs $u_{1}, u_2$ of the mapping defined by the generating matrix $\bf G$

    Figure 13.  $(u+v, v)$ polar code PE block

    Figure 14.  Blocks of the $(u+v, v)$ polar code decoders of length $N$ bits

    Figure 15.  Block diagram for the SC pipeline decoder

    Figure 16.  Block diagram for the SC line decoder

    Figure 17.  Block diagram for the limited parallelism line decoder

    Figure 18.  BP line decoder components definitions

    Figure 19.  Block diagram for the BP line decoder. Details of figure appear in Figures 20, 21 and 22 corresponding to sub-figures A, B and C respectively.

    Figure 20.  Block diagram for the BP line decoder (Figure 19) -zoom-in: Sub-figure A

    Figure 21.  Block diagram for the BP line decoder (Figure 19) -zoom-in: Sub-figure B

    Figure 22.  Block diagram for the BP line decoder (Figure 19) -zoom-in: Sub-figure C

    Figure 23.  Block definitions of SC line decoder for polar code of length $N$ based on a linear $\ell$ dimensions kernel with alphabet $F$

    Table 1.  Routing tables for OP-MUX and OP-DEMUX in Figure 18

    $c^{(opMux)}, c^{(opDeMux)}$ $c^{(BPPE)}$ $\mu^{(in)}_0$ $\mu^{(in)}_1$ $\mu^{(out)}$ Equation
    $0$ $0$ $\mu^{(in)}_{x_1}$ $\mu^{(in)}_{u_1}$ $\mu_{e_1\rightarrow a_0}$ (45)
    $1$ $1$ $\mu^{(in)}_{x_0}$ $\mu^{(in)}_{u_0}$ $\mu_{a_0 \rightarrow e_1}$ (46)
    $2$ $1$ $\mu^{(in)}_{x_0}$ $\mu_{e_1\rightarrow a_0}$ $\mu_{u_0}^{(out)}$ (47)
    $3$ $0$ $\mu^{(in)}_{x_1}$ $\mu_{a_0\rightarrow e_1}$ $\mu_{u_1}^{(out)}$ (48)
    $4$ $1$ $\mu^{(in)}_{u_0}$ $\mu_{e_1\rightarrow a_0}$ $\mu_{x_0}^{(out)}$ (49)
    $5$ $0$ $\mu^{(in)}_{u_1}$ $\mu_{a_0\rightarrow e_1}$ $\mu_{x_1}^{(out)}$ (50)
    $6$ $0$ or $1$ $\mu^{(ext, in)}_0$ $\mu^{(ext, in)}_1$ $\mu^{(ext, out)}$ (80)
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