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Applicable Algebra in Engineering, Communication and Computing

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05-01-2021 | Original Paper

Minimal PD-sets for codes associated with the graphs $Q^m_2$, m even

Authors: J. D. Key, B. G. Rodrigues

Published in: Applicable Algebra in Engineering, Communication and Computing | Issue 1/2023

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Abstract

For $m\ge 4$ even, the duals of p-ary codes, for any prime p, from adjacency matrices for the m-ary 2-cube $Q^m_2$ are shown to have subcodes with parameters $[m^2,2m-2,m]$ for which minimal PD-sets of size $\frac{m}{2}$ are constructed, hence attaining the full error-correction capabilities of the code, and, as such, the most efficient sets for full permutation decoding.

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Professor H.-J. Kroll has shown us a shorter, more compact, proof that the given set provides for full error correction.

Assmus, Jr, E. F., Key, J. D.: Designs and their codes. Cambridge University Press, Cambridge: 1992, Cambridge Tracts in Mathematics, Vol. 103 (Second printing with corrections, 1993)

Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: The user language. J. Symb. Comput. 24, 3/4, 235–265 (1997)

Cannon, J., Steel, A., White, G.: Linear codes over finite fields, Handbook of Magma Functions. In: Cannon, J., Bosma, W. (eds.) Computational Algebra Group, Department of Mathematics, University of Sydney, V2.13, http://magma.maths.usyd.edu.au/magma, pp. 3951–4023 (2006)

Daniel, M.: Gordon, Minimal permutation sets for decoding the binary Golay codes. IEEE Trans. Inform. Theory 28, 541–543 (1982)MathSciNetCrossRefMATH

Day, K., Ayyoub, A.E.A.: Fault diameter of $k$-ary $n$-cube networks. IEEE Trans. Parallel Distrib. Syst. 8(9), 903–907 (1997)CrossRef

Fish, W.: Binary codes and permutation decoding sets from the graph products of cycles. Appl. Algebra Eng. Commun. Comput. 28(5), 369–389 (2017). https://doi.org/10.1007/s00200-016-0310-yMathSciNetCrossRefMATH

Fish, W., Key, J.D., Mwambene, E.: Graphs, designs and codes related to the $n$-cube. Discrete Math. 309, 3255–3269 (2009)MathSciNetCrossRefMATH

Fish, W., Key, J.D., Mwambene, E.: Binary codes of line graphs from the $n$-cube. J. Symb. Comput. 45, 800–812 (2010)CrossRefMATH

Fish, W., Key, J.D., Mwambene, E., Rodrigues, B.G.: Hamming graphs and LCD codes. J. Appl. Math. Comput. 61, 461–479 (2019). https://doi.org/10.1007/s12190-019-01259-wMathSciNetCrossRefMATH

10.

Fish, W., Key, J.D., Mwambene, E.: Partial permutation decoding for simplex codes. Adv. Math. Commun. 6, 505–516 (2012)MathSciNetCrossRefMATH

11.

Huffman, W.C.: Codes and groups. In: Pless, V.S., Huffman, W. C. (eds.) Handbook of Coding Theory, vol. 2, Part 2, Chapter 17, pp. 1345–1440. Elsevier, Amsterdam (1998)

12.

Key, J.D., McDonough, T.P., Mavron, V.C.: Partial permutation decoding for codes from finite planes. Eur. J. Combin. 26, 665–682 (2005)MathSciNetCrossRefMATH

13.

Key, J.D., McDonough, T.P., Mavron, V.C.: Improved partial permutation decoding for Reed-Muller codes. Discrete Math. 340, 722–728 (2017)MathSciNetCrossRefMATH

14.

Key, J.D., Rodrigues, B.G.: Binary codes from $m$-ary $n$-cubes ${Q}^m_n$. Adv. Math. Commun. (2020). https://doi.org/10.3934/amc.2020079CrossRefMATH

15.

Key, J. D., Seneviratne, P.: Permutation decoding for binary self-dual codes from the graph ${Q}_n$ where $n$ is even, Advances in Coding Theory and Cryptology. Shaska, T., Huffman, W.C., Joyner, D., Ustimenko, V. (eds.) vol. 2, pp. 152–159. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, Series on Coding Theory and Cryptology (2007)

16.

Kroll, H.-J., Taherian, S.-G., Vincenti, R.: Optimal antiblocking information systems for the binary codes related to triangular graphs. Adv. Math. Commun. (2020)

17.

Kroll, H.-J., Vincenti, R.: PD-sets related to the codes of some classical varieties. Discrete Math. 301, 89–105 (2005)MathSciNetCrossRefMATH

18.

MacWilliams, F.J.: Permutation decoding of systematic codes. Bell System Tech. J. 43, 485–505 (1964)CrossRefMATH

19.

MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1983)MATH

20.

Schönheim, J.: On coverings. Pac. J. Math. 14, 1405–1411 (1964)CrossRefMATH

Title: Minimal PD-sets for codes associated with the graphs , m even
Authors: J. D. Key
B. G. Rodrigues
Publication date: 05-01-2021
Publisher: Springer Berlin Heidelberg
Published in: Applicable Algebra in Engineering, Communication and Computing / Issue 1/2023
Print ISSN: 0938-1279
Electronic ISSN: 1432-0622
DOI: https://doi.org/10.1007/s00200-020-00481-5

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Minimal PD-sets for codes associated with the graphs \(Q^m_2\), m even

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