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Sub-Quadratic Decoding of One-Point Hermitian Codes

Johan S. R. Nielsen, Peter Beelen
2015 IEEE Transactions on Information Theory  
We present the first two sub-quadratic complexity decoding algorithms for one-point Hermitian codes.  ...  The first is based on a fast realisation of the Guruswami-Sudan algorithm by using state-of-the-art algorithms from computer algebra for polynomial-ring matrix minimisation.  ...  CONCLUSION In this paper, we have demonstrated that decoding of one-point Hermitian codes in sub-quadratic complexity is possible: we describe two decoding algorithms, both of which are able to decode  ... 
doi:10.1109/tit.2015.2424415 fatcat:3g7fw6vqirfvllu7sy7crvdwdy

Twisted Hermitian Codes

Austin Allen, Keller Blackwell, Olivia Fiol, Rutuja Kshirsagar, Bethany Matsick, Gretchen L. Matthews, Zoe Nelson
2020 Mathematics  
Twisted Hermitian codes allow one to work over smaller alphabets than those based on Reed–Solomon codes of similar lengths.  ...  We define a family of codes called twisted Hermitian codes, which are based on Hermitian codes and inspired by the twisted Reed–Solomon codes described by Beelen, Puchinger, and Nielsen.  ...  Conflicts of Interest: The authors declare no conflict of interest.  ... 
doi:10.3390/math9010040 fatcat:opdqsw7bdvejvfbtjbu6bhuqi4

Improved Power Decoding of One-Point Hermitian Codes [article]

Sven Puchinger, Irene Bouw, Johan Rosenkilde né Nielsen
2017 arXiv   pre-print
We propose a new partial decoding algorithm for one-point Hermitian codes that can decode up to the same number of errors as the Guruswami--Sudan decoder.  ...  In addition, it promises improvements for decoding interleaved Hermitian codes.  ...  We use the description of power decoding for one-point Hermitian codes as in [4] .  ... 
arXiv:1703.07982v1 fatcat:ni2sziftovc7xffozm5bvrvtc4

Unitary space-time modulation via cayley transform

Yindi Jing, B. Hassibi
2003 IEEE Transactions on Signal Processing  
The codes are designed based on an information-theoretic criterion and lend themselves to polynomial-time (often cubic) near-maximum-likelihood decoding using a sphere decoding algorithm.  ...  This work can be viewed as a generalization, to the nonsquare case, of the Cayley codes that have been proposed for differential USTM.  ...  We know that the matrix is Hermitian, but it has rank now instead of full rank. Therefore, the entries of the lower right Hermitian sub-matrix of are uniquely determined by its other entries.  ... 
doi:10.1109/tsp.2003.818202 fatcat:diki4wo3fjdhphmeypeye7y6gy

Improved Power Decoding of Interleaved One-Point Hermitian Codes [article]

Sven Puchinger and Johan Rosenkilde and Irene Bouw
2018 arXiv   pre-print
We propose a new partial decoding algorithm for h-interleaved one-point Hermitian codes that can decode-under certain assumptions-an error of relative weight up to 1-(k+gn)^h/h+1, where k is the dimension  ...  In the special case h=1, we obtain an adaption of the improved power decoding algorithm to one-point Hermitian codes, which for all simulated parameters achieves a similar observed failure probability  ...  We use the description of power decoding for one-point Hermitian codes as in [4] .  ... 
arXiv:1801.07006v1 fatcat:pkmu3empnrb3dj7qyqxcxoh77y

Decoding of Subspace Codes, a Problem of Schubert Calculus over Finite Fields [article]

Joachim Rosenthal, Anna-Lena Trautmann
2012 arXiv   pre-print
Then we show how the problem of decoding of subspace codes used in random network coding can be formulated as a problem in Schubert calculus.  ...  In this article we first review some of the successes Schubert calculus had in the past.  ...  The points in P 5 representing the image of Grass 2 (2, F 4 ) are described by one quadratic equation (shuffle relation): u 1,2 u 3,4 + u 1,3 u 2,4 + u 1,4 u 2,3 = 0.  ... 
arXiv:1209.2887v1 fatcat:o32pcgb5pfe6pdpgzf66oq5ila

Optimal rate list decoding over bounded alphabets using algebraic-geometric codes [article]

Venkatesan Guruswami, Chaoping Xing
2020 arXiv   pre-print
The second class of codes are obtained by restricting evaluation points of an algebraic-geometric code to rational points from a subfield.  ...  We give new constructions of two classes of algebraic code families which are efficiently list decodable with small output list size from a fraction 1-R-ϵ of adversarial errors where R is the rate of the  ...  We are grateful to the anonymous reviewers for detailed and perceptive comments which led to significant improvements in the organization and presentation of the paper.  ... 
arXiv:1708.01070v2 fatcat:pg74jht3lnbn5ht7fufvlmqfq4

A new family of locally correctable codes based on degree-lifted algebraic geometry codes

Eli Ben-Sasson, Ariel Gabizon, Yohay Kaplan, Swastik Kopparty, Shubangi Saraf
2013 Proceedings of the 45th annual ACM symposium on Symposium on theory of computing - STOC '13  
Taking a concrete illustrating example, we show that degree-lifted Hermitian codes form a family of locally correctable codes over an alphabet that is significantly smaller than that obtained by Reed-Muller  ...  We describe new constructions of error correcting codes, obtained by "degree-lifting" a short algebraic geometry base-code of block-length q to a lifted-code of block-length q m , for arbitrary integer  ...  C L (G, D) is called a one point AG code if G is a one point divisor. Definition 4.8. A code automorphism is a permutation of the coordinates of a code, σ such that σ (C) = C.  ... 
doi:10.1145/2488608.2488714 dblp:conf/stoc/Ben-SassonGKKS13 fatcat:hwixwvv265aypdsm6qrmbfwnwq

Revisited Design Criteria For STBCs With Reduced Complexity ML Decoding [article]

Asma Mejri, Mohamed-Achraf Khsiba, Ghaya Rekaya-Ben Othmane
2016 arXiv   pre-print
In a second Quadratic Form approach, the Fast Sphere Decoding (FSD) complexity of linear STBCs is captured by a Hurwitz Radon Quadratic Form (HRQF) matrix based in its essence on the HR Theory.  ...  This appproach served to construct new families of codes admitting fast sphere decoding such as multi-group decodable, fast decodable, and fast-group decodable codes.  ...  A sub-class of fast decodable codes, termed Block-orthogonal codes has been proposed in [27] .  ... 
arXiv:1605.02899v1 fatcat:qgxq34aykfg4pprd77rbbwozee

The ubiquity of order domains for the construction of error control codes

John Little
2007 Advances in Mathematics of Communications  
The definition is largely motivated by the structures utilized in the Berlekamp-Massey-Sakata (BMS) decoding algorithm, with Feng-Rao majority voting for unknown syndromes, applied to one-point geometric  ...  To illustrate the potential usefulness of these observations, we present a series of new explicit examples of order domains associated to varieties with many rational points over finite fields: Hermitian  ...  The definition (see (2.1) below for the formulation we will use) is largely motivated by the structures utilized in the Berlekamp-Massey-Sakata (BMS) decoding algorithm for one-point geometric Goppa codes  ... 
doi:10.3934/amc.2007.1.151 fatcat:62gcwxj7zrce7e7fshtvtozqyi

The Ubiquity of Order Domains for the Construction of Error Control Codes [article]

John B. Little
2008 arXiv   pre-print
The definition is largely motivated by the structures utilized in the Berlekamp-Massey-Sakata (BMS) decoding algorithm, with Feng-Rao majority voting for unknown syndromes, applied to one-point geometric  ...  To illustrate the potential usefulness of these observations, we present a series of new explicit examples of order domains associated to varieties with many rational points over finite fields: Hermitian  ...  Each of those singular points blows up to a Hermitian curve on the Deligne-Lusztig variety.  ... 
arXiv:math/0304292v2 fatcat:qriegx764vdrpd63tqaijvtv7u

On the performance of concatenated convolutional code and Alamouti space-time code with noisy channel estimates and finite-depth interleaving

J. Jootar, J.R. Zeidler, J.G. Proakis
2008 IEEE Transactions on Communications  
Send comment regarding this burden estimates or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for  ...  , searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information.  ...  Characteristic function of a Hermitian quadratic form of a complex Gaussian random vector Let x be a zero-mean complex Gaussian random vector of length m and Q be a square matrix of order m, then the characteristic  ... 
doi:10.1109/tcomm.2008.060175 fatcat:7vnsvu4xmncqjdxbqeoaunb6ki

Coding Theory

Joachim Rosenthal, Mohammad Amin Shokrollahi
2007 Oberwolfach Reports  
The workshop on coding theory covered many facets of the recent research advances.  ...  Coding theory lies naturally at the intersection of a large number of disciplines in pure and applied mathematics: algebra and number theory, probability theory and statistics, communication theory, discrete  ...  His result is a one line formula for the actual minimum distance of Hermitian two-point codes and a shorter proof.  ... 
doi:10.4171/owr/2007/56 fatcat:ac2rttsrn5g2rn5la72nudiyum

Group-metric multiuser decoding

E.A. Fain, M.K. Varanasi
2001 IEEE Transactions on Communications  
We propose the new group metric (GM) soft-decision decoder for convolutionally coded synchronous multiple-access channels.  ...  The GM decoder exploits the independently operating encoders of the multiuser channel by making decoding decisions for a subset of the users, but incorporating all the multiuser information in its metrics  ...  Thus, counting the number of -dimensional quadratic forms evaluated per coded bit, we find the joint decoder to have complexity of order . B.  ... 
doi:10.1109/26.930632 fatcat:uljzebtpbvgldekoyoijhdaiia

On the decoding of algebraic-geometric codes

T. Hoholdt, R. Pellikaan
1995 IEEE Transactions on Information Theory  
This paper provides a survey of the existing literature on the decoding of algebraic-geometric codes. Definitions, theorems and cross references will be given.  ...  One might hope that the attitude towards algebraic-geometric codes, which Acknowledgement We hope that all those who have contributed to the decoding of algebraic-geometric codes recognize their fair share  ...  At the beginning of the invention of RS codes the engineering community had the impression that these codes would never be used in practice, since it used advanced mathematics such as Galois fields.  ... 
doi:10.1109/18.476214 fatcat:xlmzq64klnbs3dm3wvxhi7cgfm
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