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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

Hans-Joachim Kroll 1,, , Sayed-Ghahreman Taherian 2,3, and  Rita Vincenti 2,3,

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

* Corresponding author: Hans-Joachim Kroll

Dedicated to Professor Helmut Karzel on the occasion of his 92nd birthday.

Received  July 2020 Published  February 2022 Early access  August 2020

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.  Google Scholar

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H.-J. Kroll and R. Vincenti, Antiblocking decoding, Discrete Appl. Math., 158 (2010), 1461-1464.  doi: 10.1016/j.dam.2010.04.007.  Google Scholar

[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.  Google Scholar

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V. Pless and W. C. Huffman, Handbook of Coding Theory, Elsevier, Amsterdam, 1998. Google Scholar

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J. Schönheim, On Coverings, Pacific J. Math., 14 (1964), 1405-1411.  doi: 10.2140/pjm.1964.14.1405.  Google Scholar

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V. D. Tonchev, Combinatorial Configurations Designs, Codes, Graphs, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 40, Longman, New York, 1988.  Google Scholar

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.  Google Scholar

[2]

H.-J. Kroll and R. Vincenti, Antiblocking decoding, Discrete Appl. Math., 158 (2010), 1461-1464.  doi: 10.1016/j.dam.2010.04.007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

V. D. Tonchev, Combinatorial Configurations Designs, Codes, Graphs, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 40, Longman, New York, 1988.  Google Scholar

Figure .  The set $\{x, y\}, \; x < y $ belongs to $\mathcal A_l$ iff $l$ is placed in $(x, y)$
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