Problems of Information Transmission

, Volume 54, Issue 4, pp 301–328 | Cite as

Polar Codes with Higher-Order Memory

  • H. AfşerEmail author
  • H. Deliç
Coding Theory


We introduce a construction of a set of code sequences {Cn(m) : n ≥ 1, m ≥ 1} with memory order m and code length N(n). {Cn(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(nm), and {Cn(m)} coincides with the original polar codes when m = 1. We show that {Cn(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 Pe of {Cn(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) = ρmρm − 1 − 1. The encoding and decoding complexities of {Cn(m)} decrease with increasing m, which proves the existence of new polar coding schemes that have lower complexity than Arıkan’s construction.


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© Pleiades Publishing, Inc. 2018

Authors and Affiliations

  1. 1.Wireless Communications Laboratory, Department of Electrical and Electronics EngineeringBoğaziçi UniversityIstanbulTurkey
  2. 2.Department of Electrical and Electronics EngineeringAdana Science and Technology UniversityAdanaTurkey

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