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Open AccessOpen Access JUSTC Research Articles:Mathematics

Several new classes of generalized zero-difference balanced functions

  • College of Mathematics and Software Science, Sichuan Normal University, Chengdu, 610066, China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2015.12.007
  • Received Date: 13 May 2015
  • Accepted Date: 09 December 2015
  • Rev Recd Date: 09 December 2015
  • Publish Date: 30 December 2015
    Abstract

  • Abstract

    Zero-difference balanced(ZDB) functions were introduced by Ding in connection with construction of optimal constant composition codes and optimal and perfect difference systems of sets. Based on such functions, people have been constructed optimal constant weight codes and optimal frequency hopping sequences. Here the zero-difference balanced function was generalized to the generalized zero-difference balanced (G-ZDB) function, and based on properties of p-cyclotomic cosets, several new classes of generalized zero-difference balanced functions were constructed.

    Abstract

    Zero-difference balanced(ZDB) functions were introduced by Ding in connection with construction of optimal constant composition codes and optimal and perfect difference systems of sets. Based on such functions, people have been constructed optimal constant weight codes and optimal frequency hopping sequences. Here the zero-difference balanced function was generalized to the generalized zero-difference balanced (G-ZDB) function, and based on properties of p-cyclotomic cosets, several new classes of generalized zero-difference balanced functions were constructed.
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    • References(12)
  • [1]
    Ding C, Wang Q, Xiong M S. Three new families of zero-difference balanced functions with applications[DB/OL]. arXiv: 1312.4252.
    [2]
    Ding C. Optimal constant composition codes from zero-difference banlanced functions[J]. IEEE Trans Inform Theory, 2008, 54(12): 5 766-5 770.
    [3]
    Ding C. Optimal and perfect difference systems of sets[J]. J Combin Theory Ser A, 2009, 116(1): 109-119.
    [4]
    Ding C, Tan Y. Zero-difference balanced functions with application[J]. Journal of Statistical Theory and Practice, 2012, 6(1): 3-19.
    [5]
    Pott A, Wang Q. Difference balanced functions and their generalized difference sets[DB/OL]. arXiv: 1309.7842.
    [6]
    Ge G, Miao Y, Yao Z. Optimal frequency hopping sequences: auto- and cross-correlation properties[J]. IEEE Trans Inform Theory, 2009, 55(2): 867-879.
    [7]
    Nyberg K. Perfect nonlinear S-boxes[C]// Advances in cryptology- EUROCRYPT 91(Brighton, 1991). Berlin/ Heidelberg: Springer, 1991, 547: 378-386.
    [8]
    Wang Q, Zhou Y. Sets of zero-difference balanced functions and their applications[DB/OL]. arXiv: 1208.1878.
    [9]
    Yan S. Elementary Number Theory[M]. Berlin/ Heidelberg: Springer, 2002.
    [10]
    Feng T. A new construction of perfect nonlinear functions using Galois rings[J]. J Comb Designs, 2009, 17(3): 229-239.
    [11]
    Zha Z, Kyureghyan G M, Wang X. Perfect nonlinear binomials and their semifields[J]. Finite Fields and Their Applications, 2009, 15(2): 125-133.
    [12]
    Zhou Z, Tang X, Wu D, et al. Some new classes of zero-difference balanced functions[J]. IEEE Trans Inform Theory, 2012, 58(1): 139-145.
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    [1]
    Ding C, Wang Q, Xiong M S. Three new families of zero-difference balanced functions with applications[DB/OL]. arXiv: 1312.4252.
    [2]
    Ding C. Optimal constant composition codes from zero-difference banlanced functions[J]. IEEE Trans Inform Theory, 2008, 54(12): 5 766-5 770.
    [3]
    Ding C. Optimal and perfect difference systems of sets[J]. J Combin Theory Ser A, 2009, 116(1): 109-119.
    [4]
    Ding C, Tan Y. Zero-difference balanced functions with application[J]. Journal of Statistical Theory and Practice, 2012, 6(1): 3-19.
    [5]
    Pott A, Wang Q. Difference balanced functions and their generalized difference sets[DB/OL]. arXiv: 1309.7842.
    [6]
    Ge G, Miao Y, Yao Z. Optimal frequency hopping sequences: auto- and cross-correlation properties[J]. IEEE Trans Inform Theory, 2009, 55(2): 867-879.
    [7]
    Nyberg K. Perfect nonlinear S-boxes[C]// Advances in cryptology- EUROCRYPT 91(Brighton, 1991). Berlin/ Heidelberg: Springer, 1991, 547: 378-386.
    [8]
    Wang Q, Zhou Y. Sets of zero-difference balanced functions and their applications[DB/OL]. arXiv: 1208.1878.
    [9]
    Yan S. Elementary Number Theory[M]. Berlin/ Heidelberg: Springer, 2002.
    [10]
    Feng T. A new construction of perfect nonlinear functions using Galois rings[J]. J Comb Designs, 2009, 17(3): 229-239.
    [11]
    Zha Z, Kyureghyan G M, Wang X. Perfect nonlinear binomials and their semifields[J]. Finite Fields and Their Applications, 2009, 15(2): 125-133.
    [12]
    Zhou Z, Tang X, Wu D, et al. Some new classes of zero-difference balanced functions[J]. IEEE Trans Inform Theory, 2012, 58(1): 139-145.
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