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Signal-space characterization of iterative decoding

B.J. Frey, R. Koetter, A. Vardy
2001 IEEE Transactions on Information Theory  
By tracing the flow of computations in the iterative decoders for low-density parity-check codes, we formulate a signal-space view for a finite number of iterations in a finite-length code.  ...  We show that some pseudocodewords cause decoding errors, but that there are also pseudocodewords that frequently correct the deleterious effects of other pseudocodewords.  ...  This example shows that the effects of skewness and spurious pseudosignals cannot, in general, be treated independently. VI.  ... 
doi:10.1109/18.910587 fatcat:wvvel23kvfdndl22rfpgsaktlu

A 100-Gb/s-Physical-Layer Architecture for Higher-Speed Ethernet for VSR and Backplane Applications

H. TOYODA, S. NISHIMURA, M. OKUNO, M. TERADA
2007 IEICE transactions on electronics  
Ten lanes convey 320-bit data synchronously (32 bits × 10 lanes) and parity data of forward-error correction code (newly developed (544, 512) code FEC), providing highly reliable (BER<1E-22) data transmission  ...  , FEC, Fire codes Requirements for a Next-Generation Higher-Speed Ethernet Interface The required functions and specifications for nextgeneration higher-speed Ethernet are listed below.  ...  In all, 16 sets of codes, S0 to S15, can be constructed from the various combinations of five /I/s and /K/s.  ... 
doi:10.1093/ietele/e90-c.10.1957 fatcat:h5v5snircvh2zgpstyocejviwe

Decoding of Interleaved Linearized Reed-Solomon Codes with Applications to Network Coding [article]

Hannes Bartz, Sven Puchinger
2021 arXiv   pre-print
We show how to construct and decode lifted interleaved linearized Reed-Solomon codes.  ...  Compared to the construction by Martinez-Penas-Kschischang, interleaving allows to increase the decoding region significantly (especially w.r.t. the number of insertions) and decreases the overhead due  ...  In this paper we consider an interpolation-based decoding scheme in the generalized operator evaluation domain without the need for casting the decoding problem to the skew metric.  ... 
arXiv:2101.05604v4 fatcat:rkjvgtjzibhdzcrrqlsjs2kppe

Hartmann–Tzeng bound and skew cyclic codes of designed Hamming distance

José Gómez-Torrecillas, F.J. Lobillo, Gabriel Navarro, Alessando Neri
2018 Finite Fields and Their Applications  
For skew BCH codes, which are covered by our constructions, we discuss decoding algorithms. Detailed examples illustrate both the theory as the constructive methods it supports.  ...  In this paper, we give a version of the Hartmann-Tzeng bound that works for a wide class of skew cyclic codes. We also provide a practical method for constructing them with designed distance.  ...  These codes are presented as evaluation codes, and skew RS block codes are of this kind (see Remark A.12).  ... 
doi:10.1016/j.ffa.2017.11.001 fatcat:t2y6472m5zb5bhhirlierq34q4

Efficient Decoding of Folded Linearized Reed-Solomon Codes in the Sum-Rank Metric [article]

Felicitas Hörmann, Hannes Bartz
2022 arXiv   pre-print
We show how to construct h-folded linearized Reed-Solomon (FLRS) codes and derive an interpolation-based decoding scheme that is capable of correcting sum-rank errors beyond the unique decoding radius.  ...  The sum-rank analogs of Reed-Solomon and Gabidulin codes are linearized Reed-Solomon codes.  ...  The construction of folded skew Reed-Solomon codes and the transfer of the presented decoder to the skew metric are other open problems. Fig. 1 . 1 Fig. 1.  ... 
arXiv:2109.14943v2 fatcat:dd3yzncugffshmf7oq476peewi

Fast Decoding of Codes in the Rank, Subspace, and Sum-Rank Metric [article]

Hannes Bartz and Thomas Jerkovits and Sven Puchinger and Johan Rosenkilde
2021 arXiv   pre-print
problems related to the remainder-evaluation of skew polynomials.  ...  The new algorithms for the various decoding-related computational problems are interesting in their own and have further applications, in particular parts of decoders of several other codes and foundational  ...  INTRODUCTION We consider algorithms for decoding certain codes in three different metrics -rank, subspace and sum-rank metric -all of which arise as evaluation-like codes of skew polynomials.  ... 
arXiv:2005.09916v2 fatcat:sqtkw73frraqhfiol7evekjcwy

Bounds on List Decoding of Linearized Reed-Solomon Codes [article]

Sven Puchinger, Johan Rosenkilde
2021 arXiv   pre-print
List decoding in these extreme cases is well-studied, and the two code classes behave very differently in terms of list size, but nothing is known for the general case.  ...  Furthermore, we show that some families of linearized Reed-Solomon codes with constant numbers of blocks cannot be list decoded beyond the unique decoding radius.  ...  Generalized Operator Evaluation The codes studied in this paper are evaluation codes of skew polynomials, where the used evaluation map is the generalized operator evaluation [31] , defined as follows  ... 
arXiv:2102.03079v1 fatcat:3jyd2rfiivbnbhyu34fq6dc3mq

Error-Erasure Decoding of Linearized Reed-Solomon Codes in the Sum-Rank Metric [article]

Felicitas Hörmann, Hannes Bartz, Sven Puchinger
2022 arXiv   pre-print
We propose the first known error-erasure decoder for LRS codes to unleash their full potential for multishot network coding.  ...  Linearized Reed-Solomon (LRS) codes contain Reed-Solomon and Gabidulin codes as subclasses and fulfill the Singleton-like bound in the sum-rank metric with equality.  ...  Other constructions of codes in the sumrank metric include partial unit memory codes constructed from rank-metric codes [3] , [4] , convolutional codes [5] , [6] and variable block-size constructions  ... 
arXiv:2202.06758v1 fatcat:r3yru5aixvbbhcp2dmukda646u

Skew Constacyclic Codes over Finite Fields and Finite Chain Rings

Hai Q. Dinh, Bac T. Nguyen, Songsak Sriboonchitta
2016 Mathematical Problems in Engineering  
Moreover, general decoding procedure for decoding skew BCH codes with designed distance and an algorithm for decoding skew BCH codes are discussed.  ...  Among other results, we also consider the Euclidean and Hermitian dual codes of skewΘ-cyclic and skewΘ-negacyclic codes over finite chain rings in general and overFpm+uFpmin particular.  ...  Acknowledgments The authors are grateful for the Department of Mathematical Sciences, Kent State University; Faculty of Economics, Chiang Mai University; and Department of Mathematics, Mahidol University  ... 
doi:10.1155/2016/3965789 fatcat:rsc44ah5ofahzgxobvegkyjwza

Approximate Bacon-Shor code and holography

ChunJun Cao, Brad Lackey
2021 Journal of High Energy Physics  
We then construct approximate versions of the holographic hybrid codes by "skewing" the code subspace, where the size of skewing is analogous to the size of the gravitational constant in holography.  ...  This code admits a local log-depth encoding/decoding circuit, and can be represented as a holographic tensor network which satisfies an analog of the Ryu-Takayanagi formula and reproduces features of the  ...  Classification of the skewed codes Although we provided a few examples of skewed Bacon-Shor codes and the perfect state, the construction is generally applicable for other codes as well.  ... 
doi:10.1007/jhep05(2021)127 fatcat:afgrjjfqejcybp7zury4jpinqy

WDM coding for high-capacity lightwave systems

Shou-Kuo Shao, Ming-Seng Kao
1994 Journal of Lightwave Technology  
The system performance was evaluated and the system limitation imposed by bit-skew among wavelength channels was analyzed.  ...  and a channel span of 30 nm is assumed  ...  Here we evaluate the limitation of the WDM coding system imposed by bit-skew by considering three types of fibers.  ... 
doi:10.1109/50.265746 fatcat:mjjtuue5drbbfd7d24gh2vlioi

Approximate Bacon-Shor Code and Holography [article]

ChunJun Cao, Brad Lackey
2021 arXiv   pre-print
We then construct approximate versions of the holographic hybrid codes by "skewing" the code subspace, where the size of skewing is analogous to the size of the gravitational constant in holography.  ...  This code admits a local log-depth encoding/decoding circuit, and can be represented as a holographic tensor network which satisfies an analog of the Ryu-Takayanagi formula and reproduces features of the  ...  Classification of the Skewed Codes Although we provided a few examples of skewed Bacon-Shor codes and the perfect state, the construction is generally applicable for other codes as well.  ... 
arXiv:2010.05960v2 fatcat:fompftv2ivgjvecsgl5h6zdr7i

Reliable and Secure Multishot Network Coding using Linearized Reed-Solomon Codes

Umberto Martinez-Penas, Frank R. Kschischang
2019 IEEE Transactions on Information Theory  
to ρ packets, and wire-tap up to μ links, all throughout ℓ shots of a linearly-coded network.  ...  Assuming no knowledge of the underlying linear network code (in particular, the network topology and underlying linear code may be random and change with time), a coding scheme achieving zero-error communication  ...  V-C Linearized RS [36] O(n ′4 ℓ 2 log(ℓ) 2 ) TABLE II II MAXIMUM-DISTANCE EVALUATION CODES USING SKEW AND LINEARIZED POLYNOMIALS Code Metric Type of polynomials evaluated Evaluation points Lengths  ... 
doi:10.1109/tit.2019.2912165 fatcat:qziaieilgvevddundhpyqpg6ty

Foreword

Daniel Augot, Jean-Charles Faugère, Ludovic Perret
2009 Journal of symbolic computation  
Boucher and Ulmer have followed the trend of constructing good codes.  ...  In Coding Theory, there are two major goals: construction of good codes, that is to say codes which both are large and have a large correction capacity (minimum distance); the second goal is to design  ... 
doi:10.1016/j.jsc.2008.11.004 fatcat:w3feqvkbjjdlliud772hp2zkva

A Comparison of Skewed and Orthogonal Lattices in Gaussian Wiretap Channels [article]

Alex Karrila, Camilla Hollanti
2014 arXiv   pre-print
in terms of both the legitimate receiver's and the eavesdropper's probabilities.  ...  We consider lattice coset-coded transmissions over a wiretap channel with additive white Gaussian noise (AWGN).  ...  CONCLUSIONS In the construction of lattice codes for AWGN wiretap channels, skewed lattices should be taken more seriously.  ... 
arXiv:1411.5861v1 fatcat:vexxaxku4nf7pb6kgzgnuijgyu
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