On the Construction and Decoding of Concatenated Polar Codes

A scheme for concatenating the recently invented polar codes with interleaved block codes is considered. By concatenating binary polar codes with interleaved Reed-Solomon codes, we prove that the proposed concatenation scheme captures the capacity-achieving property of polar codes, while having a significantly better error-decay rate. We show that for any $\epsilon > 0$, and total frame length $N$, the parameters of the scheme can be set such that the frame error probability is less than $2^{-N^{1-\epsilon}}$, while the scheme is still capacity achieving. This improves upon 2^{-N^{0.5-\eps}}, the frame error probability of Arikan's polar codes. We also propose decoding algorithms for concatenated polar codes, which significantly improve the error-rate performance at finite block lengths while preserving the low decoding complexity

An efficient length- and rate-preserving concatenation of polar and repetition codes

We improve the method in \cite{Seidl:10} for increasing the finite-lengh performance of polar codes by protecting specific, less reliable symbols with simple outer repetition codes. Decoding of the scheme integrates easily in the known successive decoding algorithms for polar codes. Overall rate and block length remain unchanged, the decoding complexity is at most doubled. A comparison to related methods for performance improvement of polar codes is drawn.Comment: to be presented at International Zurich Seminar (IZS) 201

Scalable Successive-Cancellation Hardware Decoder for Polar Codes

Polar codes, discovered by Ar{\i}kan, are the first error-correcting codes with an explicit construction to provably achieve channel capacity, asymptotically. However, their error-correction performance at finite lengths tends to be lower than existing capacity-approaching schemes. Using the successive-cancellation algorithm, polar decoders can be designed for very long codes, with low hardware complexity, leveraging the regular structure of such codes. We present an architecture and an implementation of a scalable hardware decoder based on this algorithm. This design is shown to scale to code lengths of up to N = 2^20 on an Altera Stratix IV FPGA, limited almost exclusively by the amount of available SRAM

Bhattacharyya parameter of monomials codes for the Binary Erasure Channel: from pointwise to average reliability

Monomial codes were recently equipped with partial order relations, fact that allowed researchers to discover structural properties and efficient algorithm for constructing polar codes. Here, we refine the existing order relations in the particular case of Binary Erasure Channel. The new order relation takes us closer to the ultimate order relation induced by the pointwise evaluation of the Bhattacharyya parameter of the synthetic channels. The best we can hope for is still a partial order relation. To overcome this issue we appeal to related technique from network theory. Reliability network theory was recently used in the context of polar coding and more generally in connection with decreasing monomial codes. In this article, we investigate how the concept of average reliability is applied for polar codes designed for the binary erasure channel. Instead of minimizing the error probability of the synthetic channels, for a particular value of the erasure parameter p, our codes minimize the average error probability of the synthetic channels. By means of basic network theory results we determine a closed formula for the average reliability of a particular synthetic channel, that recently gain the attention of researchers.Comment: 21 pages, 5 figures, 3 tables. Submitted for possible publicatio