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Page 1440 of Mathematical Reviews Vol. , Issue 2001B
[page]
2001
Mathematical Reviews
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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
2008
Designs, Codes and Cryptography
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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]
2022
arXiv
pre-print
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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
2022
Symmetry
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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
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2017 pre-print arXiv:1709.09865v3 fatcat:s2tkkdnqwvckbhj2jwxemcseuuOther Versions
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]
2009
arXiv
pre-print
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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
2004
IEEE Transactions on Information Theory
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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
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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]
2017
Signals and Communication Technology
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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]
2014
arXiv
pre-print
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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]
2012
arXiv
pre-print
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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
2012
2012 International Symposium on Wireless Communication Systems (ISWCS)
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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
2001
IEEE Transactions on Information Theory
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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]
2016
arXiv
pre-print
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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
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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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