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SOME NEW CLASSES OF ZERO-DIFFERENCE BALANCED FUNCTIONS AND RELATED CONSTANT COMPOSITION CODES

  • Sankhadip, Roy (Department of Basic Science and Humanities University of Engineering and Management)
  • Received : 2021.03.06
  • Accepted : 2022.07.15
  • Published : 2022.11.30
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Abstract

Zero-difference balanced (ZDB) functions can be applied to many areas like optimal constant composition codes, optimal frequency hopping sequences etc. Moreover, it has been shown that the image set of some ZDB functions is a regular partial difference set, and hence provides strongly regular graphs. Besides, perfect nonlinear functions are zero-difference balanced functions. However, the converse is not true in general. In this paper, we use the decomposition of cyclotomic polynomials into irreducible factors over 𝔽p, where p is an odd prime to generalize some recent results on ZDB functions. Also we extend a result introduced by Claude et al. [3] regarding zero-difference-p-balanced functions over 𝔽pn. Eventually, we use these results to construct some optimal constant composition codes.

Keywords

Acknowledgement

I would like to thank my advisor Professor Robert Fitzgerald (retired) from department of mathematics, Southern Illinois University, Carbondale, USA for his valuable advice leading to writing this paper.

References

  1. G. T. Bogdanova and S. N. Kapralov, Enumeration of optimal ternary constantcomposition codes, Probl. Inf. Transm. 39 (2003), no. 4, 346-351; translated from Problemy Peredachi Informatsii 39 (2003), no. 4, 35-40. https://doi.org/10.1023/B:PRIT.0000011273.98799.a8
  2. H. Cai, X. Zeng, T. Helleseth, X. Tang, and Y. Yang, A new construction of zerodifference balanced functions and its applications, IEEE Trans. Inform. Theory 59 (2013), no. 8, 5008-5015. https://doi.org/10.1109/TIT.2013.2255114
  3. C. Carlet, G. Gong, and Y. Tan, Quadratic zero-difference balanced functions, APN functions and strongly regular graphs, Des. Codes Cryptogr. 78 (2016), no. 3, 629-654. https://doi.org/10.1007/s10623-014-0022-x
  4. Y. M. Chee, A. C. H. Ling, S. Ling, and H. Shen, The PBD-closure of constantcomposition codes, IEEE Trans. Inform. Theory 53 (2007), no. 8, 2685-2692. https://doi.org/10.1109/TIT.2007.901175
  5. W. Chu, C. J. Colbourn, and P. Dukes, Tables for constant composition codes, J. Combin. Math. Combin. Comput. 54 (2005), 57-65.
  6. W. Chu, C. J. Colbourn, and P. Dukes, On constant composition codes, Discrete Appl. Math. 154 (2006), no. 6, 912-929. https://doi.org/10.1016/j.dam.2005.09.009
  7. C. J. Colbourn, T. Klove, and A. C. H. Ling, Permutation arrays for powerline communication and mutually orthogonal Latin squares, IEEE Trans. Inform. Theory 50 (2004), no. 6, 1289-1291. https://doi.org/10.1109/TIT.2004.828150
  8. C. Ding, Optimal constant composition codes from zero-difference balanced functions, IEEE Trans. Inform. Theory 54 (2008), no. 12, 5766-5770. https://doi.org/10.1109/TIT.2008.2006420
  9. C. Ding, Optimal and perfect difference systems of sets, J. Combin. Theory Ser. A 116 (2009), no. 1, 109-119. https://doi.org/10.1016/j.jcta.2008.05.007
  10. C. Ding and Y. Tan, Zero-difference balanced functions with applications, J. Stat. Theory Pract. 6 (2012), no. 1, 3-19. https://doi.org/10.1080/15598608.2012.647479
  11. C. Ding, Q. Wang, and M. Xiong, Three new families of zero-difference balanced functions with applications, IEEE Trans. Inform. Theory 60 (2014), no. 4, 2407-2413. https://doi.org/10.1109/TIT.2014.2306821
  12. C. Ding and J. Yin, Algebraic constructions of constant composition codes, IEEE Trans. Inform. Theory 51 (2005), no. 4, 1585-1589. https://doi.org/10.1109/TIT.2005.844087
  13. C. Ding and J. Yin, Combinatorial constructions of optimal constant-composition codes, IEEE Trans. Inform. Theory 51 (2005), no. 10, 3671-3674. https://doi.org/10.1109/TIT.2005.855612
  14. L. Jiang and Q. Liao, Generalized zero-difference balanced functions and their applications, Chinese J. Contemp. Math. 37 (2016), no. 3, 201-216; translated from Chinese Ann. Math. Ser. A 37 (2016), no. 3, 243-260.
  15. L. Jiang and Q. Liao, On generalized zero-difference balanced functions, Commun. Korean Math. Soc. 31 (2016), no. 1, 41-52. https://doi.org/10.4134/CKMS.2016.31.1.041
  16. J. Knauer and J. Richstein, The continuing search for Wieferich primes, Math. Comp. 74 (2005), no. 251, 1559-1563. https://doi.org/10.1090/S0025-5718-05-01723-0
  17. R. Lidl and H. Niederreiter, Finite fields, second edition, Encyclopedia of Mathematics and its Applications, 20, Cambridge University Press, Cambridge, 1997.
  18. H. Liu and Q. Liao, Some new constructions for generalized zero-difference balanced functions, Internat. J. Found. Comput. Sci. 27 (2016), no. 8, 897-908. https://doi.org/10.1142/S0129054116500362
  19. Y. Luo, F. Fu, A. J. H. Vinck, and W. Chen, On constant-composition codes over Zq, IEEE Trans. Inform. Theory 49 (2003), no. 11, 3010-3016. https://doi.org/10.1109/TIT.2003.819339
  20. W. Meidl and A. Topuzo˘glu, Quadratic functions with prescribed spectra, Des. Codes Cryptogr. 66 (2013), no. 1-3, 257-273. https://doi.org/10.1007/s10623-012-9690-6
  21. V. Pless, Introduction to the Theory of Error-Correcting Codes, second edition, WileyInterscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., New York, 1989.
  22. L.-Z. Shen, J.-J. Wen, and F.-W. Fu, A new class of zero-difference balanced functions, Inform. Process. Lett. 136 (2018), 9-11. https://doi.org/10.1016/j.ipl.2018.03.011
  23. Z. Yi, Z. Lin, and L. Ke, A generic method to construct zero-difference balanced functions, Cryptogr. Commun. 10 (2018), no. 4, 591-609. https://doi.org/10.1007/12095-017-0247-4
  24. Z. Zha and L. Hu, Cyclotomic constructions of zero-difference balanced functions with applications, IEEE Trans. Inform. Theory 61 (2015), no. 3, 1491-1495. https://doi.org/10.1109/TIT.2014.2388231
  25. Z. Zhou, X. Tang, D. Wu, and Y. Yang, Some new classes of zero-difference balanced functions, IEEE Trans. Inform. Theory 58 (2012), no. 1, 139-145. https://doi.org/10.1109/TIT.2011.2171418