Filters








2,600 Hits in 8.9 sec

A Matrix Ring Description for Cyclic Convolutional Codes [article]

Heide Gluesing-Luerssen, Fai-Lung Tsang
2007 arXiv   pre-print
In this paper, we study convolutional codes with a specific cyclic structure. By definition, these codes are left ideals in a certain skew polynomial ring.  ...  Using that the skew polynomial ring is isomorphic to a matrix ring we can describe the algebraic parameters of the codes in a more accessible way.  ...  Besides the recently introduced classes of Goppa convolutional codes [1, 17] and group convolutional codes [2] , the main class of such codes are cyclic convolutional codes.  ... 
arXiv:0708.1343v1 fatcat:7tqalejovjb2pmm4qebbj32khu

A matrix ring description for cyclic convolutional codes

Fai-Lung Tsang, Heide Gluesing-Luerssen
2008 Advances in Mathematics of Communications  
In this paper, we study convolutional codes with a specific cyclic structure. By definition, these codes are left ideals in a certain skew polynomial ring.  ...  Using that the skew polynomial ring is isomorphic to a matrix ring we can describe the algebraic parameters of the codes in a more accessible way.  ...  Besides the recently introduced classes of Goppa convolutional codes [1, 17] and group convolutional codes [2] , the main class of such codes are cyclic convolutional codes.  ... 
doi:10.3934/amc.2008.2.55 fatcat:gpraywhug5duzkbsc5hp2go5ci

On the Parameters of Convolutional Codes with Cyclic Structure [article]

Heide Gluesing-Luerssen, Barbara Langfeld
2003 arXiv   pre-print
In this paper convolutional codes with cyclic structure will be investigated. These codes can be understood as left principal ideals in a suitable skew-polynomial ring.  ...  A complete characterization and construction will be given for minimal cyclic codes. It is derived from a detailed investigation of the units in the skew-polynomial ring.  ...  Cyclic Convolutional Codes In this section we will introduce the notion of cyclicity for convolutional codes.  ... 
arXiv:math/0312092v1 fatcat:rmnkhvsrezgmzoi4nw323udvja

Convolutional codes from units in matrix and group rings [article]

Ted Hurley
2007 arXiv   pre-print
A general method for constructing convolutional codes from units in Laurent series over matrix rings is presented.  ...  Using group ring as matrix rings, this forms a basis for in-depth exploration of convolutional codes from group ring encoding, wherein the ring in the group ring is itself a group ring.  ...  Algebraic Description of Convolutional Codes Background on general algebra and group rings may be obtained in [9] .  ... 
arXiv:0711.3629v1 fatcat:c7hixzjhwbbfth572ytjidll4a

ON THE ALGEBRAIC PARAMETERS OF CONVOLUTIONAL CODES WITH CYCLIC STRUCTURE

HEIDE GLUESING-LUERSSEN, BARBARA LANGFELD
2006 Journal of Algebra and its Applications  
In this paper convolutional codes with cyclic structure will be investigated. These codes can be understood as left principal ideals in a suitable skew-polynomial ring.  ...  A complete characterization and construction will be given for minimal cyclic codes. It is derived from a detailed investigation of the units in the skew-polynomial ring.  ...  Acknowledgement We wish to thank the anonymous reviewers for their helpful comments. They improved the exposition considerably.  ... 
doi:10.1142/s0219498806001570 fatcat:szj6gnbldrd5jovjxtkdpeo554

Page 10582 of Mathematical Reviews Vol. , Issue 2004m [page]

2004 Mathematical Reviews  
Summary: “Nested convolutional codes are a set of convolutional codes derived from a given generator matrix.  ...  The class of quasi-cyclic codes has proved to be a good source for such codes. In this paper, we use the algebraic structure of quasi- cyclic codes and the BCH type bound introduced in [K.  ... 

Page 744 of Mathematical Reviews Vol. , Issue 90A [page]

1990 Mathematical Reviews  
He presents the slid- ing groups (which are an important tool in the description of the automorphism group of a convolutional code) and a method for computing these automorphisms.  ...  In Chapter 4 the author defines the convolutional codes as a generalization of cyclic block codes by introducing the concept of automorphism.  ... 

Page 5271 of Mathematical Reviews Vol. , Issue 2000g [page]

2000 Mathematical Reviews  
A spectral representation of the cyclic codes over R is given and a BCH-like bound is given for the Lee distance of the codes.  ...  Summary: “We give a simple decoding algorithm to decode linear cyclic codes of odd length over the ring R = F)+uF = {0,1,u,ua= u+1}, where u? =0.  ... 

Distance bounds for convolutional codes and some optimal codes [article]

Heide Gluesing-Luerssen, Wiland Schmale
2003 arXiv   pre-print
Most of the examples in this paper are cyclic convolutional codes in a generalized sense as it has been introduced in the seventies.  ...  After a discussion of the Griesmer and Heller bound for the distance of a convolutional code we present several codes with various parameters, over various fields, and meeting the given distance bounds  ...  The examples are presented via a generator matrix so that no knowledge about cyclicity for convolutional codes is required from the reader.  ... 
arXiv:math/0305135v1 fatcat:f7yhrdqp2rghhmycheantr62pq

Dual skew codes from annihilators: Transpose Hamming ring extensions [article]

José Gómez-Torrecillas and F. J. Lobillo and Gabriel Navarro
2018 arXiv   pre-print
In this paper a framework to study the dual of skew cyclic codes is proposed.  ...  Our construction is applied to left ideal convolutional codes, skew constacyclic codes and skew Reed-Solomon code, showing that the dual of these codes belong to the same class.  ...  Left ideal convolutional codes of automorphism type. Let A be a ring and σ : AA a ring automorphism.  ... 
arXiv:1803.00239v1 fatcat:gmb5yrne7feqjh7w7x5ynd3rpa

BCH convolutional codes

J. Rosenthal, F.V. York
1999 IEEE Transactions on Information Theory  
Using a new parity-check matrix, a class of convolutional codes with a designed free distance is introduced.  ...  This new class of codes has many characteristics of BCH block codes, therefore, we call these codes BCH convolutional codes.  ...  Glüsing-Lüerssen, A. Loeliger, R. Smarandache, and P. Weiner for helpful discussions during the preparation of this paper. Helpful comments by the anonymous referees are acknowledged as well.  ... 
doi:10.1109/18.782104 fatcat:wqv2chligfdq7o6lx6t4o6ilfm

Page 3404 of Mathematical Reviews Vol. , Issue 98E [page]

1998 Mathematical Reviews  
The material in the paper is related to finding an encoder (generator matrix) G and inverter (left-inverse for G) L for such a code given a spanning set or a syndrome former (parity-check matrix) H over  ...  i.e. codes whose generator matrix has columns belong- ing to the points of a projective space except for the points of a certain disjoint union of subspaces.  ... 

Decoding Reed-Solomon Skew-Differential Codes [article]

José Gómez-Torrecillas, Gabriel Navarro, José Patricio Sánchez-Hernández
2020 arXiv   pre-print
A final section locates the Reed-Solomon skew-differential codes introduced here within the general context of codes defined by means of skew polynomial rings.  ...  Thus, the first part of the paper develops a direct presentation of the codes by means of parity-check matrices, and the decoding algorithm rests upon matrix and linear maps manipulations.  ...  Indeed, for some purposes, it is useful to understand the ring R as a factor ring of a ring of skew polynomials. Let us derive such a description.  ... 
arXiv:2006.00562v1 fatcat:moqt5m7xmzgzhn6woqw2gub5ta

On Doubly-Cyclic Convolutional Codes

Heide Gluesing-Luerssen, Wiland Schmale
2006 Applicable Algebra in Engineering, Communication and Computing  
Cyclicity of a convolutional code (CC) is relying on a nontrivial automorphism of the algebra F[x]/(x n − 1), where F is a finite field.  ...  A particular choice of the data leads to the class of doubly-cyclic CC's. Within this large class Reed-Solomon and BCH convolutional codes can be defined.  ...  Acknowledgements We would like to thank the reviewers for their helpful comments. References  ... 
doi:10.1007/s00200-006-0014-9 fatcat:dbrc4i5wpnfmtlkr4cobwqgaze

Number Theoretic Transforms for Secure Signal Processing [article]

Alberto Pedrouzo-Ulloa, Juan Ramón Troncoso-Pastoriza, Fernando Pérez-González
2018 arXiv   pre-print
, matrix-based processing or error correcting codes.  ...  This usage of NTTs paired with appropriate signal pre- and post-coding enables a whole range of easily composable signal processing operations comprising, among others, filtering, generalized convolutions  ...  Below, we include an example of the use of cyclic codes for the reduction of the cipher expansion in RLWE or m-RLWE based cryptosystems. a) Cyclic codes for better cipher expansion: In [22] , the authors  ... 
arXiv:1607.05229v2 fatcat:mrlorcxrrrg7lbfr7khn7mr2xq
« Previous Showing results 1 — 15 out of 2,600 results