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A New Permutation Decoding Method for Reed-Muller Codes [article]

Mikhail Kamenev, Yulia Kameneva, Oleg Kurmaev, Alexey Maevskiy
2019 arXiv   pre-print
A novel permutation decoding method for Reed-Muller codes is presented.  ...  It is demonstrated that the proposed decoding technique can take advantage of several early termination methods leading to a significant reduction of the operations number required for the decoding, with  ...  INTRODUCTION Reed-Muller (RM) codes are a family of error correcting codes discovered by Muller [1] and shortly after by Reed [2] , who also proposed the first efficient decoding algorithm.  ... 
arXiv:1901.04433v1 fatcat:jhxua2jocza3vilwb7ij2imb3q

Algebraic properties of polar codes from a new polynomial formalism

Magali Bardet, Vlad Dragoi, Ayoub Otmani, Jean-Pierre Tillich
2016 2016 IEEE International Symposium on Information Theory (ISIT)  
A polar code of length 2 m is also generated by evaluation of monomials, but not necessarily by the same monomials as a Reed-Muller code: if we want a polar code of a certain dimension for a certain channel  ...  INTRODUCTION Polar codes and Reed Muller codes viewed as monomial codes.  ...  Although the permutation group of a Reed-Muller code is well-known, the question remains open for decreasing monomial codes.  ... 
doi:10.1109/isit.2016.7541295 dblp:conf/isit/BardetDOT16 fatcat:arxg2fgemzhzlkxmqwntpvg3jm

Polar Codes for Automorphism Ensemble Decoding [article]

Charles Pillet, Valerio Bioglio, Ingmar Land
2021 arXiv   pre-print
Moreover, state-of-the-art BP-based permutation decoding for polar codes is beaten by BP-based AE thanks to this design.  ...  To overcome this problem, we provide methods to determine whether a given polar code has non-LTA automorphisms and to identify such automorphisms.  ...  Recently, a new type of permutation group decoder for Reed-Muller codes [9] , [10] , termed automorphism ensemble (AE) decoder, has been proposed based on the automorphism group of the code [11] .  ... 
arXiv:2102.08250v2 fatcat:26anwln75jbnlexelxtbcc3hq4

Reed-Muller codes achieve capacity on erasure channels

Shrinivas Kudekar, Santhosh Kumar, Marco Mondelli, Henry D. Pfister, Eren Şaşoğlu, Rüdiger Urbanke
2016 Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2016  
We introduce a new approach to proving that a sequence of deterministic linear codes achieves capacity on an erasure channel under maximum a posteriori decoding.  ...  An important consequence of this result is that a sequence of Reed-Muller codes with increasing block length and converging rate achieves capacity.  ...  Reed-Muller Codes Reed-Muller codes were introduced by Muller in [58] and, soon after, Reed proposed a majority logic decoder in [60] .  ... 
doi:10.1145/2897518.2897584 dblp:conf/stoc/KudekarKMPSU16 fatcat:buqks4llzrebzhc7pvp63ydkk4

Page 5402 of Mathematical Reviews Vol. , Issue 86k [page]

1986 Mathematical Reviews  
Author summary: “The wise use of first order Reed-Muller codes (RM-1) in communications is determinated by their optimality and by the existence of fast algorithms for their decoding.  ...  Author summary: “A new proposed automatic equalizer including a decoder of error-correcting code (AEDEC) is obtained by the joint adaptive operation of an automatic equalizer and a decoder.  ... 

Partially symmetric monomial codes [article]

Kirill Ivanov, Rüdiger Urbanke
2021 arXiv   pre-print
We introduce a new family of codes, partially symmetric monomial codes. These codes have a smaller group of symmetries than the Reed-Muller codes and are in this sense "between" RM and polar codes.  ...  Reed-Muller codes have a large automorphism group but their low-complexity maximum likelihood decoding still remains an open problem.  ...  One can now ask whether it would be possible to find codes that have a smaller group of symmetries than Reed-Muller codes but also require a smaller decoding complexity for near-ML performance.  ... 
arXiv:2001.03790v3 fatcat:gfpmenebxfhf3bhvxeux7gvlfm

Algebraic Properties of Polar Codes From a New Polynomial Formalism [article]

Magali Bardet and Vlad Dragoi and Ayoub Otmani and Jean-Pierre Tillich
2016 arXiv   pre-print
However, finding the right set of generating monomials for a polar code which optimises the decoding performances is a hard task and channel dependent.  ...  These codes are closely related to Reed-Muller codes because both can be described with the same algebraic formalism, namely they are generated by evaluations of monomials.  ...  Basically Dumer's decoding algorithm is the successive cancellation decoder of Arıkan but the performance of the decoder is much worse in this case because the choice of monomials for a Reed-Muller code  ... 
arXiv:1601.06215v2 fatcat:lekukkihhfeyxms2mdzfrpdmye

Page 4266 of Mathematical Reviews Vol. , Issue 94g [page]

1994 Mathematical Reviews  
The known result that the automorphism group of a binary Reed-Muller (RM) code is precisely G(m, 2) is generalized by the authors to show that a g-ary RM code of length q”, i.e. a gen- eralized Reed-Muller  ...  The efficacy of the new method is demonstrated by a table containing 33 new unit-memory con- volutional codes (n,k) with 5< k < 8, rates R=k/n between 4 and $ and complete memory (M =k), as well as 12  ... 

Page 2341 of Mathematical Reviews Vol. , Issue 93d [page]

1993 Mathematical Reviews  
In keeping with the general philosophy, a great deal of time is de- voted to decoding methods, algorithms and their justification: for instance, Hadamard transform decoding of first-order Reed-Muller codes  ...  ; Reed-Muller and Preparata codes.  ... 

Permutation Decoding of Polar Codes [article]

Mikhail Kamenev, Yulia Kameneva, Oleg Kurmaev, Alexey Maevskiy
2019 arXiv   pre-print
A new permutation decoding approach for polar codes is presented.  ...  It is shown that the error correction performance of the algorithm is similar to that of the SCL decoder for polar codes.  ...  In the paper, a new permutation decoding method for polar codes is proposed.  ... 
arXiv:1901.05459v1 fatcat:ttb4oh2gy5amhn5fn2d4yifqea

Classification of Automorphisms for the Decoding of Polar Codes [article]

Charles Pillet, Valerio Bioglio, Ingmar Land
2021 arXiv   pre-print
This paper proposes new polar code design principles for the low-latency automorphism ensemble (AE) decoding.  ...  Moreover, we prove that some automorphisms are redundant under AE decoding, and we propose a new automorphisms classification based on equivalence classes.  ...  The last block size s t represents the symmetry of the code [15] ; this value is n for Reed-Muller codes and usually 1 for polar codes.  ... 
arXiv:2110.14438v1 fatcat:zkjkc525ezduxb4lq2xnrf35ae

Reed-Muller Codes Achieve Capacity on Erasure Channels [article]

Santhosh Kumar, Henry D. Pfister
2015 arXiv   pre-print
This paper introduces a new approach to proving that a sequence of deterministic linear codes achieves capacity on an erasure channel under maximum a posteriori decoding.  ...  An important consequence of this result is that a sequence of Reed-Muller codes with increasing blocklength achieves capacity if its code rate converges to a number between 0 and 1.  ...  Reed-Muller codes were introduced by Muller in [17] and, soon after, Reed proposed a majority logic decoder in [18] .  ... 
arXiv:1505.05123v2 fatcat:4u4elpxgijd3he4jnkhd4pchre

Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations [article]

Seyyed Ali Hashemi, Nghia Doan, Marco Mondelli, Warren J. Gross
2018 arXiv   pre-print
Reed-Muller (RM) and polar codes are a class of capacity-achieving channel coding schemes with the same factor graph representation.  ...  Low-complexity decoding algorithms fall short in providing a good error-correction performance for RM and polar codes.  ...  RM Codes B. Polar Codes V. CONCLUSION In this paper, we first derived the total number of factor graph permutations for Reed-Muller (RM) and polar codes.  ... 
arXiv:1807.03912v1 fatcat:nqa27rm5hzhkbg44dvyd22xs2u

Reed-Muller Codes Achieve Capacity on Erasure Channels [article]

Shrinivas Kudekar, Santhosh Kumar, Marco Mondelli, Henry D. Pfister, Eren Şaşoğlu, Rüdiger Urbanke
2016 arXiv   pre-print
We introduce a new approach to proving that a sequence of deterministic linear codes achieves capacity on an erasure channel under maximum a posteriori decoding.  ...  An important consequence of this result is that a sequence of Reed-Muller codes with increasing blocklength and converging rate achieves capacity.  ...  Reed-Muller Codes Reed-Muller codes were introduced by Muller in [17] and, soon after, Reed proposed a majority logic decoder in [18] .  ... 
arXiv:1601.04689v1 fatcat:6v5u2jc7sndprb2yfq663moy5u

Quantum Fourier sampling, Code Equivalence, and the quantum security of the McEliece and Sidelnikov cryptosystems [article]

Hang Dinh, Cristopher Moore, Alexander Russell
2011 arXiv   pre-print
The Code Equivalence problem is that of determining whether two given linear codes are equivalent to each other up to a permutation of the coordinates.  ...  We revisit the claims of our previous article in the light of these classical attacks, and discuss the particular case of the Sidelnikov cryptosystem, which is based on Reed-Muller codes.  ...  We are grateful to Kirill Morozov and Nicolas Sendrier for discussions about the security of the Sidelnikov cryptosystem and methods for oblivious transfer with McEliece-type cryptosystems [Kobara et  ... 
arXiv:1111.4382v1 fatcat:ke6ggl25afa27mbpivvqrl7a7i
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