Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1310.7159

Using concatenated algebraic geometry codes in channel polarization

Abdulla Eid, Iwan Duursma

Published Oct 27, 2013• 3 citations
arXivHTMLPDF

Abstract

Polar codes were introduced by Arikan in 2008 and are the first family of error-correcting codes achieving the symmetric capacity of an arbitrary binary-input discrete memoryless channel under low complexity encoding and using an efficient successive cancellation decoding strategy. Recently, non-binary polar codes have been studied, in which one can use different algebraic geometry codes to achieve better error decoding probability. In this paper, we study the performance of binary polar codes that are obtained from non-binary algebraic geometry codes using concatenation. For binary polar codes (i.e. binary kernels) of a given length $n$, we compare numerically the use of short algebraic geometry codes over large fields versus long algebraic geometry codes over small fields. We find that for each $n$ there is an optimal choice. For binary kernels of size up to $n \leq 1,800$ a concatenated Reed-Solomon code outperforms other choices. For larger kernel sizes concatenated Hermitian codes or Suzuki codes will do better.

DOI
10.48550/arxiv.1310.7159

Related papers

Non-binary polar codes using Reed-Solomon codes and algebraic geometry codes

Ryuhei Mori, Toshiyuki Tanaka

Polar codes, introduced by Arıkan, achieve symmetric capacity of any discrete memoryless channels under low encoding and decoding complexity. Recently, non-binary polar codes have been investigated. In this paper, we calculate error probability of non-binary polar codes constructed on the basis of Reed-Solomon matrices by numerical simulations. It is confirmed that 4-ary polar codes have significantly better performance than binary polar codes on binary-input AWGN channel. We also discuss an interpretation of polar codes in terms of algebraic geometry codes, and further show that polar codes using Hermitian codes have asymptotically good performance.

arXiv:1007.3661•IEEE Information Theory Workshop (ITW)•Aug 1, 2010•94 citations

Polar Codes: Characterization of Exponent, Bounds, and Constructions

Satish Babu Korada, Eren Şaşoğlu, Rudiger L. Urbanke

Polar codes were recently introduced by Arıkan. They achieve the capacity of arbitrary symmetric binary-input discrete memoryless channels under a low complexity successive cancellation decoding strategy. The original polar code construction is closely related to the recursive construction of Reed-Muller codes and is based on the $2 \times 2$ matrix $\bigl[ 1 &0 1& 1 \bigr]$. It was shown by Arıkan and Telatar that this construction achieves an error exponent of $\frac12$, i.e., that for sufficiently large blocklengths the error probability decays exponentially in the square root of the length. It was already mentioned by Arıkan that in principle larger matrices can be used to construct polar codes. A fundamental question then is to see whether there exist matrices with exponent exceeding $\frac12$. We first show that any $\ell \times \ell$ matrix none of whose column permutations is upper triangular polarizes symmetric channels. We then characterize the exponent of a given square matrix and derive upper and lower bounds on achievable exponents. Using these bounds we show that there are no matrices of size less than 15 with exponents exceeding $\frac12$. Further, we give a general construction based on BCH codes which for large $n$ achieves exponents arbitrarily close to 1 and which exceeds $\frac12$ for size 16.

arXiv:0901.0536•Jan 5, 2009•30 citations

Attaining Capacity with Algebraic Geometry Codes through the $(U|U+V)$ Construction and Koetter-Vardy Soft Decoding

Irene Márquez-Corbella, Jean–Pierre Tillich

In this paper we show how to attain the capacity of discrete symmetric channels with polynomial time decoding complexity by considering iterated $(U|U+V)$ constructions with Reed-Solomon code or algebraic geometry code components. These codes are decoded with a recursive computation of the {\em a posteriori} probabilities of the code symbols together with the Koetter-Vardy soft decoder used for decoding the code components in polynomial time. We show that when the number of levels of the iterated $(U|U+V)$ construction tends to infinity, we attain the capacity of any discrete symmetric channel in this way. This result follows from the polarization theorem together with a simple lemma explaining how the Koetter-Vardy decoder behaves for Reed-Solomon codes of rate close to $1$. However, even if this way of attaining the capacity of a symmetric channel is essentially the Arıkan polarization theorem, there are some differences with standard polar codes. Indeed, with this strategy we can operate succesfully close to channel capacity even with a small number of levels of the iterated $(U|U+V)$ construction and the probability of error decays quasi-exponentially with the codelength in such a case (i.e. exponentially if we forget about the logarithmic terms in the exponent). We can even improve on this result by considering the algebraic geometry codes constructed in \cite{TVZ82}. In such a case, the probability of error decays exponentially in the codelength for any rate below the capacity of the channel. Moreover, when comparing this strategy to Reed-Solomon codes (or more generally algebraic geometry codes) decoded with the Koetter-Vardy decoding algorithm, it does not only improve the noise level that the code can tolerate, it also results in a significant complexity gain.

arXiv:1701.07112•Jan 24, 2017•0 citations

Convolutional Polar Codes

Andrew J. Ferris, Christoph Hirche, David Poulin

Arikan's Polar codes attracted much attention as the first efficiently decodable and capacity achieving codes. Furthermore, Polar codes exhibit an exponentially decreasing block error probability with an asymptotic error exponent upper bounded by 1/2. Since their discovery, many attempts have been made to improve the error exponent and the finite block-length performance, while keeping the bloc-structured kernel. Recently, two of us introduced a new family of efficiently decodable error-correction codes based on a recently discovered efficiently-contractible tensor network family in quantum many-body physics, called branching MERA. These codes, called branching MERA codes, include Polar codes and also extend them in a non-trivial way by substituting the bloc-structured kernel by a convolutional structure. Here, we perform an in-depth study of a particular example that can be thought of as a direct extension to Arikan's Polar code, which we therefore name Convolutional Polar codes. We prove that these codes polarize and exponentially suppress the channel's error probability, with an asymptotic error exponent log_2(3)/2 which is provably better than for Polar codes under successive cancellation decoding. We also perform finite block-size numerical simulations which display improved error-correcting capability with only a minor impact on decoding complexity.

arXiv:1704.00715•Apr 3, 2017•17 citations

Multi-Kernel Construction of Polar Codes

Frédéric Gabry, BIOGLIO V, LAND I +1 more

We propose a generalized construction for binary polar codes based on mixing multiple kernels of different sizes in order to construct polar codes of block lengths that are not only powers of integers. This results in a multi-kernel polar code with very good performance while the encoding complexity remains low and the decoding follows the same general structure as for the original Arikan polar codes. The construction provides numerous practical advantages as more code lengths can be achieved without puncturing or shortening. We observe numerically that the error-rate performance of our construction outperforms

•Jan 1, 2017•0 citations

Polar Codes: Finite Length Implementation, Error Correlations and Multilevel Modulation

Mani Bastani Parizi

Shannon, in his seminal work, formalized the transmission of data over a communication channel and determined its fundamental limits. He characterized the relation between communication rate and error probability and showed that as long as the communication rate is below the capacity of the channel, error probability can be made as small as desirable by using appropriate coding over the communication channel and letting the codeword length approach infinity. He provided the formula for capacity of discrete memoryless channel. However, his proposed coding scheme was too complex to be practical in communication systems. Polar codes, recently introduced by Arikan, are the first practical codes that are known to achieve the capacity for a large class of channel and have low encoding and decoding complexity. The original polar codes of Arikan achieve a block error probability decaying exponentially in the square root of the block length as it goes to infinity. However, it is interesting to investigate their performance in finite length as this is the case in all practical communication schemes. In this dissertation, after a brief overview on polar codes, we introduce a practical framework for simulation of error correcting codes in general. We introduce the importance sampling concept to efficiently evaluate the performance of polar codes with finite bock length. Next, based on simulation results, we investigate the performance of different genie aided decoders to mitigate the poor performance of polar codes in low to moderate block length and propose single-error correction methods to improve the performance dramatically in expense of complexity of decoder. In this context, we also study the correlation between error events in a successive cancellation decoder. Finally, we investigate the performance of polar codes in non-binary channels. We compare the code construction of Sasoglu for Q-ary channels and classical multilevel codes. We construct multilevel polar codes for Q-ary channels and provide a thorough comparison of complexity and performance of two methods in finite length.

•Jan 1, 2012•4 citations
Polar codes were introduced by Arikan in 2008 and are the first family of error-correcting codes achieving the symmetric capacity of an arbitrary binary-input discrete memoryless channel under low complexity encoding and using an efficient successive cancellation decoding strategy. Recently, non-binary polar codes have been studied, in which one can use different algebraic geometry codes to achieve better error decoding probability. In this paper, we study the performance of binary polar codes that are obtained from non-binary algebraic geometry codes using concatenation. For binary polar codes (i.e. binary kernels) of a given length nnn, we compare numerically the use of short algebraic geometry codes over large fields versus long algebraic geometry codes over small fields. We find that for each nnn there is an optimal choice. For binary kernels of size up to n≤1,800n \leq 1,800n≤1,800 a concatenated Reed-Solomon code outperforms other choices. For larger kernel sizes concatenated Hermitian codes or Suzuki codes will do better.
Polar codes, introduced by Arıkan, achieve symmetric capacity of any discrete memoryless channels under low encoding and decoding complexity. Recently, non-binary polar codes have been investigated. In this paper, we calculate error probability of non-binary polar codes constructed on the basis of Reed-Solomon matrices by numerical simulations. It is confirmed that 4-ary polar codes have significantly better performance than binary polar codes on binary-input AWGN channel. We also discuss an interpretation of polar codes in terms of algebraic geometry codes, and further show that polar codes using Hermitian codes have asymptotically good performance.
Polar codes were recently introduced by Arıkan. They achieve the capacity of arbitrary symmetric binary-input discrete memoryless channels under a low complexity successive cancellation decoding strategy. The original polar code construction is closely related to the recursive construction of Reed-Muller codes and is based on the 2×22 \times 22×2 matrix $\bigl[ 1 &0 1& 1 \bigr]$. It was shown by Arıkan and Telatar that this construction achieves an error exponent of 12\frac1221​, i.e., that for sufficiently large blocklengths the error probability decays exponentially in the square root of the length. It was already mentioned by Arıkan that in principle larger matrices can be used to construct polar codes. A fundamental question then is to see whether there exist matrices with exponent exceeding 12\frac1221​. We first show that any ℓ×ℓ\ell \times \ellℓ×ℓ matrix none of whose column permutations is upper triangular polarizes symmetric channels. We then characterize the exponent of a given square matrix and derive upper and lower bounds on achievable exponents. Using these bounds we show that there are no matrices of size less than 15 with exponents exceeding 12\frac1221​. Further, we give a general construction based on BCH codes which for large nnn achieves exponents arbitrarily close to 1 and which exceeds 12\frac1221​ for size 16.
Attaining Capacity with Algebraic Geometry Codes through the (U∣U+V)(U|U+V)(U∣U+V) Construction and Koetter-Vardy Soft Decoding
In this paper we show how to attain the capacity of discrete symmetric channels with polynomial time decoding complexity by considering iterated (U∣U+V)(U|U+V)(U∣U+V) constructions with Reed-Solomon code or algebraic geometry code components. These codes are decoded with a recursive computation of the {\em a posteriori} probabilities of the code symbols together with the Koetter-Vardy soft decoder used for decoding the code components in polynomial time. We show that when the number of levels of the iterated (U∣U+V)(U|U+V)(U∣U+V) construction tends to infinity, we attain the capacity of any discrete symmetric channel in this way. This result follows from the polarization theorem together with a simple lemma explaining how the Koetter-Vardy decoder behaves for Reed-Solomon codes of rate close to 111. However, even if this way of attaining the capacity of a symmetric channel is essentially the Arıkan polarization theorem, there are some differences with standard polar codes. Indeed, with this strategy we can operate succesfully close to channel capacity even with a small number of levels of the iterated (U∣U+V)(U|U+V)(U∣U+V) construction and the probability of error decays quasi-exponentially with the codelength in such a case (i.e. exponentially if we forget about the logarithmic terms in the exponent). We can even improve on this result by considering the algebraic geometry codes constructed in \cite{TVZ82}. In such a case, the probability of error decays exponentially in the codelength for any rate below the capacity of the channel. Moreover, when comparing this strategy to Reed-Solomon codes (or more generally algebraic geometry codes) decoded with the Koetter-Vardy decoding algorithm, it does not only improve the noise level that the code can tolerate, it also results in a significant complexity gain.
Arikan's Polar codes attracted much attention as the first efficiently decodable and capacity achieving codes. Furthermore, Polar codes exhibit an exponentially decreasing block error probability with an asymptotic error exponent upper bounded by 1/2. Since their discovery, many attempts have been made to improve the error exponent and the finite block-length performance, while keeping the bloc-structured kernel. Recently, two of us introduced a new family of efficiently decodable error-correction codes based on a recently discovered efficiently-contractible tensor network family in quantum many-body physics, called branching MERA. These codes, called branching MERA codes, include Polar codes and also extend them in a non-trivial way by substituting the bloc-structured kernel by a convolutional structure. Here, we perform an in-depth study of a particular example that can be thought of as a direct extension to Arikan's Polar code, which we therefore name Convolutional Polar codes. We prove that these codes polarize and exponentially suppress the channel's error probability, with an asymptotic error exponent log_2(3)/2 which is provably better than for Polar codes under successive cancellation decoding. We also perform finite block-size numerical simulations which display improved error-correcting capability with only a minor impact on decoding complexity.
We propose a generalized construction for binary polar codes based on mixing multiple kernels of different sizes in order to construct polar codes of block lengths that are not only powers of integers. This results in a multi-kernel polar code with very good performance while the encoding complexity remains low and the decoding follows the same general structure as for the original Arikan polar codes. The construction provides numerous practical advantages as more code lengths can be achieved without puncturing or shortening. We observe numerically that the error-rate performance of our construction outperforms
Shannon, in his seminal work, formalized the transmission of data over a communication channel and determined its fundamental limits. He characterized the relation between communication rate and error probability and showed that as long as the communication rate is below the capacity of the channel, error probability can be made as small as desirable by using appropriate coding over the communication channel and letting the codeword length approach infinity. He provided the formula for capacity of discrete memoryless channel. However, his proposed coding scheme was too complex to be practical in communication systems. Polar codes, recently introduced by Arikan, are the first practical codes that are known to achieve the capacity for a large class of channel and have low encoding and decoding complexity. The original polar codes of Arikan achieve a block error probability decaying exponentially in the square root of the block length as it goes to infinity. However, it is interesting to investigate their performance in finite length as this is the case in all practical communication schemes. In this dissertation, after a brief overview on polar codes, we introduce a practical framework for simulation of error correcting codes in general. We introduce the importance sampling concept to efficiently evaluate the performance of polar codes with finite bock length. Next, based on simulation results, we investigate the performance of different genie aided decoders to mitigate the poor performance of polar codes in low to moderate block length and propose single-error correction methods to improve the performance dramatically in expense of complexity of decoder. In this context, we also study the correlation between error events in a successive cancellation decoder. Finally, we investigate the performance of polar codes in non-binary channels. We compare the code construction of Sasoglu for Q-ary channels and classical multilevel codes. We construct multilevel polar codes for Q-ary channels and provide a thorough comparison of complexity and performance of two methods in finite length.