We introduce a construction of a set of code sequences {C_n^(m) : n ≥ 1, m ≥ 1} with memory order m and code length N(n). {C_n^(m)} is a generalization of polar codes presented by Arıkan in [1], where the encoder mapping with length N(n) is obtained recursively from the encoder mappings with lengths N(n − 1) and N(n − m), and {C_n^(m)} coincides with the original polar codes when m = 1. We show that {C_n^(m)} achieves the symmetric capacity I(W) of an arbitrary binary-input, discrete-output memoryless channel W for any fixed m. We also obtain an upper bound on the probability of block-decoding error P_e of {C_n^(m)} and show that \({P_e} = O({2^{ - {N^\beta }}})\) is achievable for β < 1/[1+m(ϕ − 1)], where ϕ ∈ (1, 2] is the largest real root of the polynomial F(m, ρ) = ρ^m − ρ^{m − 1} − 1. The encoding and decoding complexities of {C_n^(m)} decrease with increasing m, which proves the existence of new polar coding schemes that have lower complexity than Arıkan’s construction.

中文翻译：

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具有高阶记忆的Polar码

我们引入了组码序列的结构{ Ç _Ñ^（米）：Ñ ≥1，米≥1}与存储器顺序米和码长Ñ（Ñ）。{ C _n^（m） }是Arıkan在[1]中提出的极性码的一般化，其中从长度为N（n − 1）和N（n的编码器映射中递归获得长度为N（n）的编码器映射。− m）和{ C _n^（米） }与原来的极性一致的代码时米= 1我们表明，{ Ç _Ñ^（米） }达到对称容量我（ w ^）任意的二进制输入，离散输出无记忆信道的W ^对于任何固定米。我们还获得{ C _n^（m） }的块解码错误概率P _e的上限，并证明\（{P_e} = O（{2 ^ {-{N ^ \ beta}}}）\ ）对于β <1 / [1+ m（ ϕ − 1）]是可实现的，其中ϕ∈（1,2]是多项式的最大实根˚F（米，ρ）= ρ^米- ρ^{米- 1} - 1编码和解码{的复杂Ç _Ñ^（米） }减小随米，这证明了存在比Arıkan的结构复杂度更低的新极性编码方案。