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

IEEE International Symposium on Information Theory (ISIT)

Constructing polar codes for non-binary alphabets and MACs

Ido Tal, Artyom Sharov, Alexander Vardy

Published Jul 1, 2012• 40 citations
Publisher

Abstract

Consider a channel with an input alphabet that is finite but not necessarily binary. A method for approximating such a channel having a large output alphabet size by a degraded version of it having a smaller output alphabet size is presented and analyzed. The approximation method is used to construct polar codes for both single-user and multiple-access channels with prime input alphabet sizes.

DOI
10.1109/isit.2012.6283739

Related papers

Upgraded Approximation of Non-Binary Alphabets for Polar Code Construction

Arash Ghayoori, Thomas Aaron Gulliver

An algorithm is presented for approximating a single-user channel with a prime input alphabet size. The result is an upgraded version of the channel with a reduced output alphabet size. It is shown that this algorithm can be used to reduce the output alphabet size to the input alphabet size in most cases.

arXiv:1304.1790•Apr 5, 2013•8 citations

Improved Construction for Multiplicative Repetition Based Non-Binary Polar Codes

Rongchi Xu, Peiyao Chen, Ling Liu +2 more

Conventional construction of non-binary polar codes divides the synthesized channels into the frozen channels and information channels. Each information channel carries one symbol i.e.$q$bits. However, there are many middle channels with insufficient polarization, which can not carry one symbol of$q$bits but only$i$bits,$1 \leq i<q, i \in \mathbb{Z}$, at finite block lengths. In this paper, we consider the bit-level construction for multiplicative repetition (MR) based non-binary polar codes and propose bitlevel improved construction for channel degradation method and Monte-Carlo method. Simulation results show that the proposed construction can significantly improve the decoding performance compared with the conventional construction scheme.

IEEE Information Theory Workshop (ITW)•Nov 24, 2024•1 citation

Finite-Length Scaling for Polar Codes

Hamed Hassani, Kasra Alishahi, Rudiger L. Urbanke

Consider a binary-input memoryless output-symmetric channel\(W\). Such a channel has a capacity, call it\(I(W)\), and for any\(R<I(W)\)and strictly positive constant\(P_{\rm e}\)we know that we can construct a coding scheme that allows transmission at rate\(R\)with an error probability not exceeding\(P_{\rm e}\). Assume now that we let the rate\(R\)tend to\(I(W)\)and we ask how we have to scale the blocklength\(N\)in order to keep the error probability fixed to\(P_{\rm e}\). We refer to this as the finite-length scaling behavior. This question was addressed by Strassen as well as Polyanskiy, Poor, and Verdu, and the result is that\(N\)must grow at least as the square of the reciprocal of\(I(W)-R\). Polar codes are optimal in the sense that they achieve capacity. In this paper, we are asking to what degree they are also optimal in terms of their finite-length behavior. Since the exact scaling behavior depends on the choice of the channel, our objective is to provide scaling laws that hold universally for all binary-input memoryless output-symmetric channels. Our approach is based on analyzing the dynamics of the un-polarized channels. More precisely, we provide bounds on (the exponent of) the number of subchannels whose Bhattacharyya constant falls in a fixed interval\([a,b]\). Mathematically, this can be stated as bounding the sequence\(\bigl \{{1}/{n} \log {\rm {Pr}}(Z_{n} \!\in \! [a,b])\bigr \}_{n \in \mathbb {N}}\), where\(Z_{n}\)is the Bhattacharyya process. We then use these bounds to derive tradeoffs between the rate and the block-length. The main results of this paper can be summarized as follows. Consider the sum of Bhattacharyya parameters of subchannels chosen (by the polar coding scheme) to transmit information. If we require this sum to be smaller than a given value\(P_{\rm e}>0\), then the required block-length\(N\)scales in terms of the rate\(R < I(W)\)as\(N \geq {\alpha }/{(I(W)-R)^{\underline {\mu }}}\), where\(\alpha \)is a positive constant that depends on\(P_{\rm e}\)and\(I(W)\). We show that\(\underline {\mu } = 3.579\)is a valid choice, and we conjecture that indeed the value of\(\underline {\mu }\)can be improved to\(\underline {\mu }=3.627\), the parameter for the binary erasure channel. Also, we show that with the same requirement on the sum of Bhattacharyya parameters, the block-length scales in terms of the rate like\(N \leq {\beta }/{(I(W)-R)^{\overline {\mu }}}\), where\(\beta \)is a constant that depends on\(P_{\rm e}\)and\(I(W)\), and\(\overline {\mu }=6\).

arXiv:1304.4778•IEEE Transactions on Information Theory•Jul 25, 2014•108 citations

Polar Codes for Arbitrary Classical-Quantum Channels and Arbitrary cq-MACs

Rajai Nasser, Joseph M. Renes

We prove polarization theorems for arbitrary classical-quantum (cq) channels. The input alphabet is endowed with an arbitrary Abelian group operation, and an Arıkan-style transformation is applied using this operation. It is shown that as the number of polarization steps becomes large, the synthetic cq-channels polarize to deterministic homomorphism channels that project their input to a quotient group of the input alphabet. This result is used to construct polar codes for arbitrary cq-channels and arbitrary cq multiple access channels. The encoder can be implemented inO(NlogN) operations, whereNis the blocklength of the code. A quantum successive cancellation decoder for the constructed codes is proposed. It is shown that the probability of error of this decoder decays faster than 2(-N)βfor any β <; (1/2).

arXiv:1701.03397•IEEE Transactions on Information Theory•Sep 21, 2018•11 citations

Finite-length scaling of polar codes

Hamed Hassani, Kasra Alishahi, Rudiger L. Urbanke

Consider a binary-input memoryless output-symmetric channel $W$. Such a channel has a capacity, call it $I(W)$, and for any $R<I(W)$ and strictly positive constant $P_{\rm e}$ we know that we can construct a coding scheme that allows transmission at rate $R$ with an error probability not exceeding $P_{\rm e}$. Assume now that we let the rate $R$ tend to $I(W)$ and we ask how we have to "scale" the blocklength $N$ in order to keep the error probability fixed to $P_{\rm e}$. We refer to this as the "finite-length scaling" behavior. This question was addressed by Strassen as well as Polyanskiy, Poor and Verdu, and the result is that $N$ must grow at least as the square of the reciprocal of $I(W)-R$. Polar codes are optimal in the sense that they achieve capacity. In this paper, we are asking to what degree they are also optimal in terms of their finite-length behavior. Our approach is based on analyzing the dynamics of the un-polarized channels. The main results of this paper can be summarized as follows. Consider the sum of Bhattacharyya parameters of sub-channels chosen (by the polar coding scheme) to transmit information. If we require this sum to be smaller than a given value $P_{\rm e}>0$, then the required block-length $N$ scales in terms of the rate $R < I(W)$ as $N \geq \frac{\alpha}{(I(W)-R)^{\underline{\mu}}}$, where $\alpha$ is a positive constant that depends on $P_{\rm e}$ and $I(W)$, and $\underline{\mu} = 3.579$. Also, we show that with the same requirement on the sum of Bhattacharyya parameters, the block-length scales in terms of the rate like $N \leq \frac{\beta}{(I(W)-R)^{\overline{\mu}}}$, where $\beta$ is a constant that depends on $P_{\rm e}$ and $I(W)$, and $\overline{\mu}=6$.

•Aug 18, 2016•130 citations

Polar codes for arbitrary classical-quantum channels and arbitrary cq-MACs

Rajai Nasser, Joseph M. Renes

We prove polarization theorems for arbitrary classical-quantum (cq) channels. The input alphabet is endowed with an arbitrary Abelian group operation and an Arikan-style transformation is applied using this operation. It is shown that as the number of polarization steps becomes large, the synthetic cq-channels polarize to deterministic homomorphism channels that project their input to a quotient group of the input alphabet. This result is used to construct polar codes for arbitrary cq-channels and arbitrary classical-quantum multiple access channels (cq-MAC). The encoder can be implemented in O(N log N) operations, where N is the blocklength of the code. A quantum successive cancellation decoder for the constructed codes is proposed. It is shown that the probability of error of this decoder decays faster than 2-Nβfor any β <; ½.

IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2017•3 citations
Consider a channel with an input alphabet that is finite but not necessarily binary. A method for approximating such a channel having a large output alphabet size by a degraded version of it having a smaller output alphabet size is presented and analyzed. The approximation method is used to construct polar codes for both single-user and multiple-access channels with prime input alphabet sizes.
Conventional construction of non-binary polar codes divides the synthesized channels into the frozen channels and information channels. Each information channel carries one symbol i.e.qqqbits. However, there are many middle channels with insufficient polarization, which can not carry one symbol ofqqqbits but onlyiiibits,1≤i<q,i∈Z1 \leq i<q, i \in \mathbb{Z}1≤i<q,i∈Z, at finite block lengths. In this paper, we consider the bit-level construction for multiplicative repetition (MR) based non-binary polar codes and propose bitlevel improved construction for channel degradation method and Monte-Carlo method. Simulation results show that the proposed construction can significantly improve the decoding performance compared with the conventional construction scheme.
Consider a binary-input memoryless output-symmetric channelWWW. Such a channel has a capacity, call itI(W)I(W)I(W), and for anyR<I(W)R<I(W)R<I(W)and strictly positive constantPeP_{\rm e}Pe​we know that we can construct a coding scheme that allows transmission at rateRRRwith an error probability not exceedingPeP_{\rm e}Pe​. Assume now that we let the rateRRRtend toI(W)I(W)I(W)and we ask how we have to scale the blocklengthNNNin order to keep the error probability fixed toPeP_{\rm e}Pe​. We refer to this as the finite-length scaling behavior. This question was addressed by Strassen as well as Polyanskiy, Poor, and Verdu, and the result is thatNNNmust grow at least as the square of the reciprocal ofI(W)−RI(W)-RI(W)−R. Polar codes are optimal in the sense that they achieve capacity. In this paper, we are asking to what degree they are also optimal in terms of their finite-length behavior. Since the exact scaling behavior depends on the choice of the channel, our objective is to provide scaling laws that hold universally for all binary-input memoryless output-symmetric channels. Our approach is based on analyzing the dynamics of the un-polarized channels. More precisely, we provide bounds on (the exponent of) the number of subchannels whose Bhattacharyya constant falls in a fixed interval[a,b][a,b][a,b]. Mathematically, this can be stated as bounding the sequence{1/nlog⁡Pr(Zn ⁣∈ ⁣[a,b])}n∈N\bigl \{{1}/{n} \log {\rm {Pr}}(Z_{n} \!\in \! [a,b])\bigr \}_{n \in \mathbb {N}}{1/nlogPr(Zn​∈[a,b])}n∈N​, whereZnZ_{n}Zn​is the Bhattacharyya process. We then use these bounds to derive tradeoffs between the rate and the block-length. The main results of this paper can be summarized as follows. Consider the sum of Bhattacharyya parameters of subchannels chosen (by the polar coding scheme) to transmit information. If we require this sum to be smaller than a given valuePe>0P_{\rm e}>0Pe​>0, then the required block-lengthNNNscales in terms of the rateR<I(W)R < I(W)R<I(W)asN≥α/(I(W)−R)μ‾N \geq {\alpha }/{(I(W)-R)^{\underline {\mu }}}N≥α/(I(W)−R)μ​, whereα\alpha αis a positive constant that depends onPeP_{\rm e}Pe​andI(W)I(W)I(W). We show thatμ‾=3.579\underline {\mu } = 3.579μ​=3.579is a valid choice, and we conjecture that indeed the value ofμ‾\underline {\mu }μ​can be improved toμ‾=3.627\underline {\mu }=3.627μ​=3.627, the parameter for the binary erasure channel. Also, we show that with the same requirement on the sum of Bhattacharyya parameters, the block-length scales in terms of the rate likeN≤β/(I(W)−R)μ‾N \leq {\beta }/{(I(W)-R)^{\overline {\mu }}}N≤β/(I(W)−R)μ​, whereβ\beta βis a constant that depends onPeP_{\rm e}Pe​andI(W)I(W)I(W), andμ‾=6\overline {\mu }=6μ​=6.
We prove polarization theorems for arbitrary classical-quantum (cq) channels. The input alphabet is endowed with an arbitrary Abelian group operation, and an Arıkan-style transformation is applied using this operation. It is shown that as the number of polarization steps becomes large, the synthetic cq-channels polarize to deterministic homomorphism channels that project their input to a quotient group of the input alphabet. This result is used to construct polar codes for arbitrary cq-channels and arbitrary cq multiple access channels. The encoder can be implemented inO(NlogN) operations, whereNis the blocklength of the code. A quantum successive cancellation decoder for the constructed codes is proposed. It is shown that the probability of error of this decoder decays faster than 2(-N)βfor any β <; (1/2).
Consider a binary-input memoryless output-symmetric channel WWW. Such a channel has a capacity, call it I(W)I(W)I(W), and for any R<I(W)R<I(W)R<I(W) and strictly positive constant PeP_{\rm e}Pe​ we know that we can construct a coding scheme that allows transmission at rate RRR with an error probability not exceeding PeP_{\rm e}Pe​. Assume now that we let the rate RRR tend to I(W)I(W)I(W) and we ask how we have to "scale" the blocklength NNN in order to keep the error probability fixed to PeP_{\rm e}Pe​. We refer to this as the "finite-length scaling" behavior. This question was addressed by Strassen as well as Polyanskiy, Poor and Verdu, and the result is that NNN must grow at least as the square of the reciprocal of I(W)−RI(W)-RI(W)−R. Polar codes are optimal in the sense that they achieve capacity. In this paper, we are asking to what degree they are also optimal in terms of their finite-length behavior. Our approach is based on analyzing the dynamics of the un-polarized channels. The main results of this paper can be summarized as follows. Consider the sum of Bhattacharyya parameters of sub-channels chosen (by the polar coding scheme) to transmit information. If we require this sum to be smaller than a given value Pe>0P_{\rm e}>0Pe​>0, then the required block-length NNN scales in terms of the rate R<I(W)R < I(W)R<I(W) as N≥α(I(W)−R)μ‾N \geq \frac{\alpha}{(I(W)-R)^{\underline{\mu}}}N≥(I(W)−R)μ​α​, where α\alphaα is a positive constant that depends on PeP_{\rm e}Pe​ and I(W)I(W)I(W), and μ‾=3.579\underline{\mu} = 3.579μ​=3.579. Also, we show that with the same requirement on the sum of Bhattacharyya parameters, the block-length scales in terms of the rate like N≤β(I(W)−R)μ‾N \leq \frac{\beta}{(I(W)-R)^{\overline{\mu}}}N≤(I(W)−R)μ​β​, where β\betaβ is a constant that depends on PeP_{\rm e}Pe​ and I(W)I(W)I(W), and μ‾=6\overline{\mu}=6μ​=6.
We prove polarization theorems for arbitrary classical-quantum (cq) channels. The input alphabet is endowed with an arbitrary Abelian group operation and an Arikan-style transformation is applied using this operation. It is shown that as the number of polarization steps becomes large, the synthetic cq-channels polarize to deterministic homomorphism channels that project their input to a quotient group of the input alphabet. This result is used to construct polar codes for arbitrary cq-channels and arbitrary classical-quantum multiple access channels (cq-MAC). The encoder can be implemented in O(N log N) operations, where N is the blocklength of the code. A quantum successive cancellation decoder for the constructed codes is proposed. It is shown that the probability of error of this decoder decays faster than 2-Nβfor any β <; ½.