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

* 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.

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. KeyJ. 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.

[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. KeyJ. 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.

[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.

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