Figure .
The set $\{x, y\}, \; x < y $ belongs to $\mathcal A_l$ iff $l$ is placed in $(x, y)$
-
Previous Article
${\sf {FAST}}$: Disk encryption and beyond
- AMC Home
- This Issue
-
Next Article
Infinite families of 2-designs from two classes of binary cyclic codes with three nonzeros
February
2022, 16(1): 169-183.
doi: 10.3934/amc.2020107
Optimal antiblocking systems of information sets for the binary codes related to triangular graphs
| 1. | Zentrum Mathematik, Technische Universität München, 80290 München, Germany |
| 2. | Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, 84156-83111, Iran |
| 3. | Dipartimento di Matematica e Informatica Università degli Studi di Perugia, Via Vanvitelli 106123 Perugia, Italy |
We present AI-systems for the binary codes obtained from the adjacency relation of the triangular graphs $ T(n) $ for any $ n\ge 5 $. These AI-systems are optimal and have for $ n $ odd the full error-correcting capability.
Keywords:
Linear codes,
triangular graphs,
permutation decoding,
antiblocking decoding,
antiblocking systems.
Mathematics Subject Classification:
Primary: 94B05, 94B35.
Citation:
Hans-Joachim Kroll, Sayed-Ghahreman Taherian, Rita Vincenti. Optimal antiblocking systems of information sets for the binary codes related to triangular graphs. Advances in Mathematics of Communications,
2022, 16
(1)
: 169-183.
doi: 10.3934/amc.2020107
![]()
References:
| [1] |
J. D. Key, J. Moori and B. G. Rodrigues,
Permutation decoding for the binary codes from triangular
graphs, European J. Combin., 25 (2004), 113-123.
doi: 10.1016/j.ejc.2003.08.001. |
| [2] |
H.-J. Kroll and R. Vincenti,
Antiblocking decoding, Discrete Appl. Math., 158 (2010), 1461-1464.
doi: 10.1016/j.dam.2010.04.007. |
| [3] |
H.-J. Kroll and R. Vincenti,
How to find small AI-systems for antiblocking decoding, Discrete Math., 312 (2012), 657-665.
doi: 10.1016/j.disc.2011.06.014. |
| [4] |
V. Pless and W. C. Huffman, Handbook of Coding Theory, Elsevier, Amsterdam, 1998. Google Scholar |
| [5] |
J. Schönheim,
On Coverings, Pacific J. Math., 14 (1964), 1405-1411.
doi: 10.2140/pjm.1964.14.1405. |
| [6] |
V. D. Tonchev, Combinatorial Configurations Designs, Codes, Graphs, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 40, Longman, New York, 1988. |
show all references
References:
| [1] |
J. D. Key, J. Moori and B. G. Rodrigues,
Permutation decoding for the binary codes from triangular
graphs, European J. Combin., 25 (2004), 113-123.
doi: 10.1016/j.ejc.2003.08.001. |
| [2] |
H.-J. Kroll and R. Vincenti,
Antiblocking decoding, Discrete Appl. Math., 158 (2010), 1461-1464.
doi: 10.1016/j.dam.2010.04.007. |
| [3] |
H.-J. Kroll and R. Vincenti,
How to find small AI-systems for antiblocking decoding, Discrete Math., 312 (2012), 657-665.
doi: 10.1016/j.disc.2011.06.014. |
| [4] |
V. Pless and W. C. Huffman, Handbook of Coding Theory, Elsevier, Amsterdam, 1998. Google Scholar |
| [5] |
J. Schönheim,
On Coverings, Pacific J. Math., 14 (1964), 1405-1411.
doi: 10.2140/pjm.1964.14.1405. |
| [6] |
V. D. Tonchev, Combinatorial Configurations Designs, Codes, Graphs, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 40, Longman, New York, 1988. |
| [1] |
Washiela Fish, Jennifer D. Key, Eric Mwambene. Partial permutation decoding for simplex codes. Advances in Mathematics of Communications, 2012, 6 (4) : 505-516. doi: 10.3934/amc.2012.6.505 |
| [2] |
Kwankyu Lee. Decoding of differential AG codes. Advances in Mathematics of Communications, 2016, 10 (2) : 307-319. doi: 10.3934/amc.2016007 |
| [3] |
Elisa Gorla, Felice Manganiello, Joachim Rosenthal. An algebraic approach for decoding spread codes. Advances in Mathematics of Communications, 2012, 6 (4) : 443-466. doi: 10.3934/amc.2012.6.443 |
| [4] |
Alexander Barg, Arya Mazumdar, Gilles Zémor. Weight distribution and decoding of codes on hypergraphs. Advances in Mathematics of Communications, 2008, 2 (4) : 433-450. doi: 10.3934/amc.2008.2.433 |
| [5] |
Terasan Niyomsataya, Ali Miri, Monica Nevins. Decoding affine reflection group codes with trellises. Advances in Mathematics of Communications, 2012, 6 (4) : 385-400. doi: 10.3934/amc.2012.6.385 |
| [6] |
Heide Gluesing-Luerssen, Uwe Helmke, José Ignacio Iglesias Curto. Algebraic decoding for doubly cyclic convolutional codes. Advances in Mathematics of Communications, 2010, 4 (1) : 83-99. doi: 10.3934/amc.2010.4.83 |
| [7] |
Hannes Bartz, Antonia Wachter-Zeh. Efficient decoding of interleaved subspace and Gabidulin codes beyond their unique decoding radius using Gröbner bases. Advances in Mathematics of Communications, 2018, 12 (4) : 773-804. doi: 10.3934/amc.2018046 |
| [8] |
Joan-Josep Climent, Diego Napp, Raquel Pinto, Rita Simões. Decoding of $2$D convolutional codes over an erasure channel. Advances in Mathematics of Communications, 2016, 10 (1) : 179-193. doi: 10.3934/amc.2016.10.179 |
| [9] |
Johan Rosenkilde. Power decoding Reed-Solomon codes up to the Johnson radius. Advances in Mathematics of Communications, 2018, 12 (1) : 81-106. doi: 10.3934/amc.2018005 |
| [10] |
Irene I. Bouw, Sabine Kampf. Syndrome decoding for Hermite codes with a Sugiyama-type algorithm. Advances in Mathematics of Communications, 2012, 6 (4) : 419-442. doi: 10.3934/amc.2012.6.419 |
| [11] |
Anas Chaaban, Vladimir Sidorenko, Christian Senger. On multi-trial Forney-Kovalev decoding of concatenated codes. Advances in Mathematics of Communications, 2014, 8 (1) : 1-20. doi: 10.3934/amc.2014.8.1 |
| [12] |
Vladimir Sidorenko, Christian Senger, Martin Bossert, Victor Zyablov. Single-trial decoding of concatenated codes using fixed or adaptive erasing. Advances in Mathematics of Communications, 2010, 4 (1) : 49-60. doi: 10.3934/amc.2010.4.49 |
| [13] |
Peter Beelen, Kristian Brander. Efficient list decoding of a class of algebraic-geometry codes. Advances in Mathematics of Communications, 2010, 4 (4) : 485-518. doi: 10.3934/amc.2010.4.485 |
| [14] |
Alexey Frolov, Victor Zyablov. On the multiple threshold decoding of LDPC codes over GF(q). Advances in Mathematics of Communications, 2017, 11 (1) : 123-137. doi: 10.3934/amc.2017007 |
| [15] |
Julia Lieb, Raquel Pinto. A decoding algorithm for 2D convolutional codes over the erasure channel. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021031 |
| [16] |
Fernando Hernando, Tom Høholdt, Diego Ruano. List decoding of matrix-product codes from nested codes: An application to quasi-cyclic codes. Advances in Mathematics of Communications, 2012, 6 (3) : 259-272. doi: 10.3934/amc.2012.6.259 |
| [17] |
Robert F. Bailey, John N. Bray. Decoding the Mathieu group M12. Advances in Mathematics of Communications, 2007, 1 (4) : 477-487. doi: 10.3934/amc.2007.1.477 |
| [18] |
Anna-Lena Horlemann-Trautmann, Violetta Weger. Information set decoding in the Lee metric with applications to cryptography. Advances in Mathematics of Communications, 2021, 15 (4) : 677-699. doi: 10.3934/amc.2020089 |
| [19] |
Ahmed S. Mansour, Holger Boche, Rafael F. Schaefer. The secrecy capacity of the arbitrarily varying wiretap channel under list decoding. Advances in Mathematics of Communications, 2019, 13 (1) : 11-39. doi: 10.3934/amc.2019002 |
| [20] |
Henry Cohn, Nadia Heninger. Ideal forms of Coppersmith's theorem and Guruswami-Sudan list decoding. Advances in Mathematics of Communications, 2015, 9 (3) : 311-339. doi: 10.3934/amc.2015.9.311 |
2020 Impact Factor: 0.935
Tools
Article outline
Figures and Tables
[Back to Top]