Designs and Lattices from Classes of Cyclic Linear Ternary Codes over GF(3)

Mary Immaculate Okombo; Michael Onyango Ojiema; Benard Kivunge; Vincent Marani

doi:10.51867/scimundi.maths.5.1.11

Authors

Mary Immaculate Okombo Department of Mathematics, Masinde Muliro University of Science and Technology, P. O. Box 190-50100, Kakamega, Kenya
Michael Onyango Ojiema Department of Mathematics, Masinde Muliro University of Science and Technology, P. O. Box 190-50100, Kakamega, Kenya https://orcid.org/0000-0001-9635-7597
Benard Kivunge Department of Mathematics, Kenyatta University P.O. Box 43844-00100, Nairobi, Kenya https://orcid.org/0000-0002-4905-7783
Vincent Marani Department of Mathematics, Kibabii University, P. O. Box 1699 -50200, Bungoma, Kenya https://orcid.org/0000-0002-2170-7131

DOI:

https://doi.org/10.51867/scimundi.maths.5.1.11

Keywords:

Designs, Lattices, Ternary Cyclic Codes

Abstract

In this study, we investigate the relationships between code parameters and lattice properties, providing new insights into the structure of ternary codes from a geometric perspective. Our findings extend the existing knowledge of ternary cyclic codes, particularly for lengths exceeding 25. We construct several new codes with favorable parameters, constructed previously unreported combinatorial designs, and characterized lattices with unique properties. The results demonstrate that ternary cyclic codes exhibit high structural regularity and often produce interesting designs and lattices with properties distinct from their binary counterparts. The research reveal strong interconnections between Coding Theory, Combinatorial Design Theory, and Lattice Theory in the context of ternary codes. We provide a multifaceted characterization framework that integrates algebraic, combinatorial, and geometric perspectives, offering a holistic understanding of these codes. This study contributes to the theoretical advancement of non-binary codes and opens new avenues for their practical applications in error correction, cryptography, and communication systems.

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References

Amiri, N. (2012). Automorphism of cyclic codes. Intelligent Information Management, 4, 309-310.

https://doi.org/10.4236/iim.2012.225043 DOI: https://doi.org/10.4236/iim.2012.225043

Anderson, I. and Honkala, I. (2012). A Short Course in Combinatorial Designs. E-edition.

Bienert, R. and Klopsch, B. (2010). Automorphism groups of cyclic codes. Journal of Algebraic Combinatorics, 31, 33-52.

https://doi.org/10.1007/s10801-009-0179-y DOI: https://doi.org/10.1007/s10801-009-0179-y

Conway, J.H. and Sloane, N.J.A. (1999). Sphere Packings, Lattices and Groups. Springer-Verlag, New York.

https://doi.org/10.1007/978-1-4757-6568-7 DOI: https://doi.org/10.1007/978-1-4757-6568-7

Ding, C. and Helleseth, T. (2013). The weight distribution of some irreducible cyclic codes. IEEE Transactions on Information Theory, 59(9), 5898-5904. DOI: https://doi.org/10.1109/TIT.2013.2260795

Ding, C., Kohel, D.R., and Ling, S. (2000). Secret-sharing with a class of ternary codes. Theoretical Computer Science, 246, 285-298. https://doi.org/10.1016/S0304-3975(00)00207-3 DOI: https://doi.org/10.1016/S0304-3975(00)00207-3

Ding, C. and Wang, X. (2005). A coding theory construction of new systematic authentication codes. Theoretical Computer Science, 330, 81-99. https://doi.org/10.1016/j.tcs.2004.09.011 DOI: https://doi.org/10.1016/j.tcs.2004.09.011

Ding, C. and Zhou, Z. (2017). Parameters of 2-designs from Some BCH codes. Springer International Publishing, 10194, 110-127. https://doi.org/10.1007/978-3-319-55589-8_8 DOI: https://doi.org/10.1007/978-3-319-55589-8_8

van Eupen, M., van Lint, J.H. (1993). On the minimum distance of ternary cyclic codes. IEEE Transactions on Information Theory, 39(2), 409-422. https://doi.org/10.1109/18.212272 DOI: https://doi.org/10.1109/18.212272

Ebeling, W. (2013). Lattices and codes (pp. 1-32). Springer Fachmedien Wiesbaden. https://doi.org/10.1007/978-3-658-00360-9_1 DOI: https://doi.org/10.1007/978-3-658-00360-9_1

Fette, B., et al. (2008). RF and Wireless Technologies. Elsevier, Inc., Oxford, London.

Harada, M. and Tonchev, V.D. (2003). Self-orthogonal codes from symmetric designs with fixed-point-free automorphisms. Discrete Mathematics, 264, 81-90. https://doi.org/10.1016/S0012-365X(02)00553-8 DOI: https://doi.org/10.1016/S0012-365X(02)00553-8

Huffman, W.C. and Pless, V. (2003). Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge.

https://doi.org/10.1017/CBO9780511807077 DOI: https://doi.org/10.1017/CBO9780511807077

Kamiya, N.(2007). High-rate quasi-cyclic low-density parity-check codes derived from finite affine planes. IEEE Transactions on Information Theory, 53(4), 1444-1459.

https://doi.org/10.1109/TIT.2007.892770 DOI: https://doi.org/10.1109/TIT.2007.892770

MacKay, D.J.C. (1999). Good error-correcting codes based on very sparse matrices. IEEE Transactions on Information Theory, 45(2), 399-431. https://doi.org/10.1109/18.748992 DOI: https://doi.org/10.1109/18.748992

MacWilliams, F.J. and Sloane, N.J.A. (1977). The Theory of Error-Correcting Codes. North-Holland Publishing Company, Amsterdam.

Micciancio, D., and Regev, O. (2009). Lattice-based cryptography. In Post-quantum cryptography (pp. 147-191). Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-540-88702-7_5 DOI: https://doi.org/10.1007/978-3-540-88702-7_5

Ozbudak, E. K., Ozbudak, F., and Saygı, Z. (2011). A class of authentication codes with secrecy. Designs, Codes and Cryptography, 59, 287-318. DOI: https://doi.org/10.1007/s10623-010-9461-1

Rains, E. M., Sloane, N. J. A., and Stufken, J. (2002). The lattice of N-run orthogonal arrays. Journal of Statistical Planning and Inference, 102(2), 477-500. https://doi.org/10.1016/S0378-3758(01)00119-7 DOI: https://doi.org/10.1016/S0378-3758(01)00119-7

Strehl, A. (2004). Ternary Codes through Ternary Designs. Australasian Journal of Combinatorics, 30(1), 1-17.

Tonchev, V.D. (1989). Self-orthogonal designs and extremal doubly even codes. Journal of Combinatorial Theory, Series A, 52, 197-205. https://doi.org/10.1016/0097-3165(89)90030-7 DOI: https://doi.org/10.1016/0097-3165(89)90030-7

Xiang, Q. (2005). Recent Progress in Algebraic Design Theory. Finite Fields and Their Applications, 11, 622-653.

https://doi.org/10.1016/j.ffa.2005.06.003 DOI: https://doi.org/10.1016/j.ffa.2005.06.003