Timeline for Hot-topics in error correcting coding related to interesting math. ?
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19 events
| when toggle format | what | by | license | comment | |
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| Jan 9, 2014 at 8:04 | answer | added | Peter Dukes | timeline score: 4 | |
| Feb 18, 2013 at 9:53 | comment | added | Alexander Chervov | arxiv.org/abs/1302.3747 Construction of minimal non-abelian left group codes Gabriela Olteanu, Inneke Van Gelder Algorithms to construct minimal left group codes are provided. These are based on results describing a complete set of orthogonal primitive idempotents in each Wedderburn component of a semisimple finite group algebra FG for a large class of groups G. As an illustration of our methods, alternative constructions to some best linear codes over F_2 and F_3 are given. | |
| Jan 15, 2013 at 13:14 | comment | added | Alexander Chervov | mathoverflow.net/questions/93765/… Mézard, Montanari's book stanford.edu/~montanar/RESEARCH/book.html | |
| Jan 14, 2013 at 6:27 | comment | added | Alexander Chervov | arxiv.org/abs/1301.2479 Weight Distribution of a Class of Cyclic Codes with Arbitrary Number of Zeros Jing Yang, Maosheng Xiong, Cunsheng Ding Cyclic codes have been widely used in digital communication systems and consume electronics as they have efficient encoding and decoding algorithms. The weight distribution of cyclic codes has been an important topic of study for many years. It is in general hard to determine the weight distribution of linear codes. In this paper, a class of cyclic codes with any number of zeros are described and their weight distributions are determined. | |
| Jan 14, 2013 at 6:22 | comment | added | Alexander Chervov | arxiv.org/abs/1301.2165 List Decoding of Lifted Gabidulin Codes via the Plücker Embedding Codes in the Grassmannian have recently found an application in random network coding. All the codewords in such codes are subspaces of $\F_q^n$ with a given dimension. In this paper, we consider the problem of list decoding of a certain family of codes in the Grassmannian, called lifted Gabidulin codes. For this purpose we use the Pl\"ucker embedding of the Grassmannian. We describe a way of representing a subset of the Pl\"ucker coordinates of lifted Gabidulin codes as linear block codes. | |
| Jan 9, 2013 at 6:14 | comment | added | Alexander Chervov | Codes, Horn's problem and Gromov-Witten invariants Alberto Besana, Cristina Martinez arxiv.org/abs/1301.1652 We study the Horn problem in the context of algebraic codes on a smooth projective curve defined over a finite field, reducing the problem to the representation theory of the special linear group $SL(2,F_q)$. We characterize the coefficients that appear in the Kronecker product of symmetric functions in terms of Gromov-Witten invariants of the Hilbert scheme of points in the plane. In addition we classify all the algebraic codes defined over the rational normal curve. | |
| Jan 4, 2013 at 11:14 | answer | added | Yuichiro Fujiwara | timeline score: 5 | |
| Nov 22, 2012 at 11:16 | comment | added | Alexander Chervov | Hajime Matsui arxiv.org/abs/1211.4728 Lemma for Linear Feedback Shift Registers and DFTs Applied to Affine Variety Codes | |
| Nov 7, 2012 at 10:16 | comment | added | Alexander Chervov | How applying Myhill-Nerode methods to hypergraphs helps mastering the Art of Trellis Decoding arxiv.org/abs/1211.1299 A trellis is a graph associated with a linear code that is used for maximum-likelihood decoding. The decoding complexity of a linear code is strongly influenced by the state complexity of the trellis, which highly depends on the coordinate permutation of the linear code. The problem of finding the coordinate permutation of a linear code such that the state-complexity of the associated trellis is at most k has been referred to as the Art of Trellis Decoding and is NP-hard | |
| Oct 19, 2012 at 12:08 | comment | added | Alexander Chervov | arxiv.org/abs/1210.5189 Accurate lower bounds on two-dimensional constraint capacities from corner transfer matrices Yao-ban Chan, Andrew Rechnitzer We analyse the capacity of several two-dimensional constraint families - the exclusion, colouring, parity and charge model families. Using Baxter's corner transfer matrix formalism combined with the corner transfer matrix renormalisation group method of Nishino and Okunishi, we calculate very tight lower bounds and estimates on the growth rate of these models. Our results strongly improve previous known lower bounds, and lead to the conj | |
| Oct 15, 2012 at 5:49 | comment | added | Alexander Chervov | arxiv.org/abs/1210.3101 Unique Decoding of General AG Codes Kwankyu Lee, Maria Bras-Amorós, Michael E. O'Sullivan ........A unique decoding algorithm for general AG codes, namely multipoint evaluation codes on algebraic curves, is presented. It is a natural generalization of the previous decoding algorithm which was only for one-point AG codes. As such, it retains the same advantages of fast speed and regular structure with the previous algorithm. Compared with other known decoding algorithms for general AG codes, it is much simpler in its description and implementation. | |
| Oct 15, 2012 at 5:48 | comment | added | Alexander Chervov | arxiv.org/abs/1210.3449 Construction of Block Orthogonal STBCs and Reducing Their Sphere Decoding Complexity G. R. Jithamithra, B. Sundar Rajan ............ We also provide new construction of block orthogonal codes from Cyclic Division Algebras (CDAs) and Crossed-Product Algebras (CPAs). In addition, we show how the block orthogonal property of the STBCs can be exploited to reduce the decoding complexity of a sphere decoder using a depth first search approach. ............... | |
| Oct 2, 2012 at 5:12 | comment | added | Alexander Chervov | arxiv.org/abs/1210.0140 Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes Cyclic, negacyclic and constacyclic codes are part of a larger class of codes called polycyclic codes; namely, those codes which can be viewed as ideals of a factor ring of a polynomial ring. The structure of the ambient ring of polycyclic codes over GR(p^a,m) and generating sets for its ideals are considered. Along with some structure details of the ambient ring, the existance of a certain type of generating set for an ideal is proven. | |
| Oct 2, 2012 at 5:08 | comment | added | Alexander Chervov | arxiv.org/abs/1210.0083 Decoding a Class of Affine Variety Codes with Fast DFT Hajime Matsui An efficient procedure for error-value calculations based on fast discrete Fourier transforms (DFT) in conjunction with Berlekamp-Massey-Sakata algorithm for a class of affine variety codes is proposed. Our procedure is achieved by multidimensional DFT and linear recurrence relations from Grobner basis and is applied to erasure-and-erro ... A motivating example of our algorithm in case of a Reed-Solomon code and a numerical example of our algorithm in case of a Hermitian code are also described. | |
| Sep 22, 2012 at 17:16 | comment | added | Alexander Chervov | books.google.com/… Coding theory and algebraic curves over finite fields G VAN DER GEER - … of Algebraic Geometry to Coding Theory, …, 2001 ... tautological classes on the moduli space and the not less spectacular proof by Kontsevich of this ... Deligne and Lusztig showed that irreducible representations of finite Lie groups can be found in a ... For example, the bounds on the number of rational points on cu | |
| Sep 18, 2012 at 6:25 | comment | added | Alexander Chervov | arxiv.org/abs/1209.3460 Expander-like Codes based on Finite Projective Geometry Swadesh Choudhary, Hrishikesh Sharma, B. S. Adiga, Sachin Patkar (Submitted on 16 Sep 2012) We present a novel error correcting code and decoding algorithm which have construction similar to expander codes. The code is based on a bipartite graph derived from the subsumption relations of finite projective geometry, and Reed-Solomon codes as component codes. We use a modified version of well-known Zemor's decoding algorithm for expander codes, for decoding our codes. ........... | |
| Mar 26, 2012 at 19:33 | answer | added | Jyrki Lahtonen | timeline score: 10 | |
| Mar 25, 2012 at 19:50 | comment | added | Asaf | There is certainly some "recent" mathematical research going into ECC as applications. The first example that comes into mind are expanders graphs, and the construction of LDPC codes by Margulis (see for example here - nd.edu/~rosen/Paper/margulis_8.pdf). | |
| Mar 25, 2012 at 19:21 | history | asked | Alexander Chervov | CC BY-SA 3.0 |