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Page 1440 of Mathematical Reviews Vol. , Issue 2001B [page]

2001 Mathematical Reviews  
In this correspondence, using the structure of quasi-cyclic (QC) codes with a BCH-type bound given in [K. Lally and P.  ...  A necessary condition required from a linear code in or- der to have a linear tail-biting trellis representation that achieves 94 INFORMATION AND COMMUNICATION, CIRCUITS 1440 the square-root bound is presented  ... 

A family of ternary quasi-perfect BCH codes

Danyo Danev, Stefan Dodunekov
2008 Designs, Codes and Cryptography  
In this paper we present a family of ternary quasi-perfect BCH codes. These codes are of minimum distance 5 and covering radius 3.  ...  The first member of this family is the ternary quadratic-residue code of length 13.  ...  Obviously the dimension of C s is k = n − 2s = 3 s − 1 2 − 2s. To determine the minimum distance we use the BCH bound for the minimum distance of a cyclic code.  ... 
doi:10.1007/s10623-008-9193-7 fatcat:nbht5okvhfhshbn2cvlam2ketm

Distance bounds for generalized bicycle codes [article]

Renyu Wang, Leonid P. Pryadko
2022 arXiv   pre-print
For such GB codes with a given maximum generator weight w, we constructed upper distance bounds by mapping them to codes local in D≤ w-1 dimensions, and lower existence bounds which give d≥ O(n^1/2).  ...  In addition, low-density parity-check GB codes have a naturally overcomplete set of low-weight stabilizer generators, which is expected to improve their performance in the presence of syndrome measurement  ...  codes can beat the square root distance bound (ignoring any logarithmic corrections).  ... 
arXiv:2203.17216v1 fatcat:q6elh3qmizhv3lctb5zu5ol3dm

Distance Bounds for Generalized Bicycle Codes

Renyu Wang, Leonid P. Pryadko
2022 Symmetry  
For such GB codes with a given maximum generator weight w, we constructed upper distance bounds by mapping them to codes local in D≤w−1 dimensions, and lower existence bounds which give d≥O(n1/2).  ...  In addition, low-density parity-check GB codes have a naturally overcomplete set of low-weight stabilizer generators, which is expected to improve their performance in the presence of syndrome measurement  ...  index-two quasi-cyclic linear codes.  ... 
doi:10.3390/sym14071348 fatcat:i6zyrfycznd4lhsw6xqpgkwphy

Asymptotically Good Additive Cyclic Codes Exist

2018 IEEE Communications Letters  
Long quasi-cyclic codes of any fixed index >1 have been shown to be asymptotically good, depending on Artin primitive root conjecture in (A. Alahmadi, C. Güneri, H. Shoaib, P. Solé, 2017).  ...  We use this recent result to construct good long additive cyclic codes on any extension of fixed degree of the base field.  ...  Acknowlegement: The authors thanks Shai Evra, C. Gueneri, and Alexander Lubotzky for helpful discussions.  ... 
doi:10.1109/lcomm.2018.2863370 fatcat:ew3jbgd4vbfubpxmpj5aoorjvu

Spectral Graph Analysis of Quasi-Cyclic Codes [article]

Roxana Smarandache, Mark F. Flanagan
2009 arXiv   pre-print
In this paper we analyze the bound on the additive white Gaussian noise channel (AWGNC) pseudo-weight of a (c,d)-regular linear block code based on the two largest eigenvalues of H^T H.  ...  We also give a necessary condition for the bound to be attained for a circulant matrix H and show a few classes of cyclic codes satisfying this criterion.  ...  Spectral graph analysis was used in [5] , and more recently, in [6] , to obtain bounds on the minimum Hamming weight, and minimum AWGNC pseudo-weight, respectively, of a length-n (c, d)-regular code  ... 
arXiv:0908.1966v2 fatcat:wf27b2jr6zb6ffasgodn3qbbcu

On Existence of Good Self-Dual Quasi-Cyclic Codes

B.K. Dey
2004 IEEE Transactions on Information Theory  
8 -quasi-cyclic codes of length respectively 2 and 8 which asymptotically meet the Gilbert-Varshamov bound.  ...  When compared with the order of the defining permutation groups, these classes of codes lie between the 2-quasi-cyclic codes and the 3-quasi-cyclic codes of length , considered in previous works.  ...  When a a a is nonzero, there are three possibilities for the pair We proceed to enumerate or get an upper bound on the number of codewords a a a (of any self-dual 2p-quasi-cyclic code) of a given weight  ... 
doi:10.1109/tit.2004.831855 fatcat:ikpv23jkkvhq3gjrwayfky5gie

Page 727 of Mathematical Reviews Vol. 40, Issue 3 [page]

1970 Mathematical Reviews  
Some results on quasi-cyclic codes. Information and Control 15 (1969), 407-423. Authors’ summary : “In this paper a class of linear error- correcting block codes is investigated.  ...  Also, a short computer generated list of the best quasi- cyclic codes of rate 4 is given. The last result is a proof that there exist very long quasicyclic codes which meet the Gilbert bound [E. N.  ... 

LDPC Codes [chapter]

Martin Tomlinson, Cen Jung Tjhai, Marcel A. Ambroze, Mohammed Ahmed, Mubarak Jibril
2017 Signals and Communication Technology  
The theory of LDPC codes is related to a branch of mathematics called graph theory. Some basic definitions used in graph theory are briefly introduced as follows.  ...  Chapter 12 LDPC Codes Background and Notation LDPC codes are linear block codes whose parity-check matrix-as the name implies-is sparse.  ...  (θ(z)) . (12.4) Minimum Hamming distance bound The lower bound of the minimum Hamming distance of a cyclic code, defined by idempotent u(x), is given by its BCH bound, which is determined by the number  ... 
doi:10.1007/978-3-319-51103-0_12 fatcat:mmavpvnx55axbo2oeksl27l6w4

Design of Quantum Stabilizer Codes From Quadratic Residues Sets [article]

Yixuan Xie, Jinhong Yuan, Qifu Sun
2014 arXiv   pre-print
For the cyclic quantum stabilizer codes, we show that they meet the existing distance bounds in literature.  ...  The proposed Type-II stabilizer codes are of quasi-cyclic (QC) structure and code length N=pk, where k is the size of a quadratic residue set.  ...  A well known subclass of LDPC codes, namely quasi-cyclic LDPC codes, also possess the simplicity of encoding.  ... 
arXiv:1407.8249v1 fatcat:y5y73mq7ezc7xbnbnralisa4bu

On the decoding of quasi-BCH codes [article]

Morgan Barbier, Clément Pernet (INRIA Grenoble Rhône-Alpes / LIG Laboratoire d'Informatique de Grenoble), Guillaume Quintin
2012 arXiv   pre-print
In the first part of this paper we show that quasi-BCH codes can be derived from Reed-Solomon codes over square matrices extending the known relation about classical BCH and Reed-Solomon codes.  ...  In this paper we investigate the structure of quasi-BCH codes.  ...  We recall some basic definitions of Reed-Solomon codes over rings in this section.  ... 
arXiv:1212.5663v1 fatcat:wbqquoaegfdohc2kiwpigffb2a

Generalised Quasi-Cyclic LDPC codes based on Progressive Edge Growth Techniques for block fading channels

Andre G. D. Uchoa, Cornelius Healy, Rodrigo C. de Lamare, Richard D. Souza
2012 2012 International Symposium on Wireless Communication Systems (ISWCS)  
Generalised Quasi-Cyclic Root-Check LDPC codes based on Progressive Edge Growth (PEG) techniques for blockfading channels are proposed.  ...  The proposed PEG-based algorithm for generalised Quasi-Cyclic Root-Check LDPC codes takes into account the specific structure, and designs a parity-check matrix with a reduced number of cycles.  ...  We have figured out one way to design such Root-Check code in a structured Quasi-Cyclic LDPC code.  ... 
doi:10.1109/iswcs.2012.6328513 dblp:conf/iswcs/UchoaHLS12 fatcat:euknppurv5bthoytddlklbkpom

Low-density parity-check codes based on finite geometries: a rediscovery and new results

Y. Kou, S. Lin, M.P.C. Fossorier
2001 IEEE Transactions on Information Theory  
Furthermore, they can be put in either cyclic or quasi-cyclic form. Consequently, their encoding can be achieved in linear time and implemented with simple feedback shift registers.  ...  Four classes of LDPC codes are constructed based on the lines and points of Euclidean and projective geometries over finite fields.  ...  ACKNOWLEDGMENT The authors wish to thank the referees for their valuable and constructive comments that improved the presentation of this paper.  ... 
doi:10.1109/18.959255 fatcat:luevsonfsjbx7i33dmb3oeatgu

On self-dual double circulant codes [article]

Adel Alahmadi, Funda Özdemir, Patrick Solé
2016 arXiv   pre-print
Exact counting formulae are derived for them and used to show they contain families of codes with relative distance satisfying a modified Gilbert-Varshamov bound.  ...  Self-dual double circulant codes of odd dimension are shown to be dihedral in even characteristic and consta-dihedral in odd characteristic.  ...  Acknowledgement: The authors are indebted to Hatoon Shoaib for helpful discussions. The second author was supported by TÜBİTAK 2214-International Doctoral Research Fellowship Programme.  ... 
arXiv:1603.00762v1 fatcat:giqtvzzcijbtxglvm732orreyi

Page 7506 of Mathematical Reviews Vol. , Issue 98K [page]

1998 Mathematical Reviews  
Summary: “Eighteen new codes are presented which improve the bounds on maximum minimum distance for binary linear codes. They are rate (m—r)/pm, r > 1, degenerate quasi-cyclic (QC) codes.  ...  A new approach to determine a lower bound on the minimum distance for algebraic-geometric codes defined from a class of plane curves is introduced, based on the generalized Bezout theorem.  ... 
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