doi: 10.3934/amc.2021030
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Partitioned difference families: The storm has not yet passed

1. 

Dipartimento di Matematica e Informatica, Università di Perugia, via Vanvitelli 1, 06123 Italy

2. 

Mathematical Institute, University of Augsburg, 86135 Augsburg, Germany

* Corresponding author: Marco Buratti

Received  February 2021 Early access August 2021

Two years ago, we alarmed the scientific community about the large number of bad papers in the literature on zero difference balanced functions, where direct proofs of seemingly new results are presented in an unnecessarily lengthy and convoluted way. Indeed, these results had been proved long before and very easily in terms of difference families.

In spite of our report, papers of the same kind continue to proliferate. Regrettably, a further attempt to put the topic in order seems unavoidable. While some authors now follow our recommendation of using the terminology of partitioned difference families, their methods are still the same and their results are often trivial or even wrong. In this note, we show how a very recent paper of this type can be easily dealt with.

Citation: Marco Buratti, Dieter Jungnickel. Partitioned difference families: The storm has not yet passed. Advances in Mathematics of Communications, doi: 10.3934/amc.2021030
References:
[1]

R. J. R. Abel and M. Buratti, Difference families, in Handbook of Combinatorial Designs, Second Edition, Chapman & Hall/CRC, Boca Raton, FL, (2007), 392–410.

[2] T. BethD. Jungnickel and H. Lenz, Design Theory, Cambridge University Press, Cambridge, 1986. 
[3]

R. C. Bose and W. S. Connor, Combinatorial properties of group divisible incomplete block designs, Ann. Math. Stat., 23 (1952), 367-383.  doi: 10.1214/aoms/1177729382.

[4]

M. Buratti, On disjoint (v, k, k-1) difference families, Des. Codes Cryptogr., 87 (2019), 745-755.  doi: 10.1007/s10623-018-0511-4.

[5]

M. Buratti and D. Jungnickel, Partitioned difference families versus zero difference balanced functions, Des. Codes Cryptogr., 87 (2019), 2461-2467.  doi: 10.1007/s10623-019-00632-x.

[6]

M. Buratti, J. Yan and C. Wang, From a $1$-rotational RBIBD to a partitioned difference family, Electronic J. Combin., 17 (2010), Research Paper 139, 23pp.

[7]

C. Ding and J. Yin, Combinatorial constructions of optimal constant composition codes, IEEE Trans. Inform. Theory, 51 (2005), 3671-3674.  doi: 10.1109/TIT.2005.855612.

[8]

S. Furino, Difference families from rings, Discret. Math., 97 (1991), 177-190.  doi: 10.1016/0012-365X(91)90433-3.

[9]

D. Jungnickel, Composition theorems for difference families and regular planes, Discrete Math., 23 (1978), 151-158.  doi: 10.1016/0012-365X(78)90113-9.

[10]

D. Jungnickel, On difference matrices and regular Latin squares, Abh. Math. Sem. Univ. Hamburg, 50 (1980), 219-231.  doi: 10.1007/BF02941430.

[11]

D. Jungnickel, On automorphism groups of divisible designs, Canadian J. Math., 34 (1982), 257-297.  doi: 10.4153/CJM-1982-018-x.

[12]

D. Jungnickel, A. Pott and K. W. Smith, Difference sets, in Handbook of Combinatorial Designs, Second Edition, Chapman & Hall/CRC, Boca Raton, FL, (2006), 419–435.

[13]

S. LiH. Wei and G. Ge, Generic constructions for partitioned difference families with applications: A unified combinatorial approach, Des. Codes Cryptogr., 82 (2017), 583-599.  doi: 10.1007/s10623-016-0182-y.

[14]

S. Xu, L. Qu and X. Cao, Three classes of partitioned difference families and their optimal constant composition codes, Adv. Math. Commun., (2020). doi: 10.3934/amc.2020120.

[15]

X. ZengH. CaiX. Tang and Y. Yang, Optimal frequency hopping sequences of odd length, IEEE Trans. Inf. Theory, 59 (2013), 3237-3248.  doi: 10.1109/TIT.2013.2237754.

show all references

References:
[1]

R. J. R. Abel and M. Buratti, Difference families, in Handbook of Combinatorial Designs, Second Edition, Chapman & Hall/CRC, Boca Raton, FL, (2007), 392–410.

[2] T. BethD. Jungnickel and H. Lenz, Design Theory, Cambridge University Press, Cambridge, 1986. 
[3]

R. C. Bose and W. S. Connor, Combinatorial properties of group divisible incomplete block designs, Ann. Math. Stat., 23 (1952), 367-383.  doi: 10.1214/aoms/1177729382.

[4]

M. Buratti, On disjoint (v, k, k-1) difference families, Des. Codes Cryptogr., 87 (2019), 745-755.  doi: 10.1007/s10623-018-0511-4.

[5]

M. Buratti and D. Jungnickel, Partitioned difference families versus zero difference balanced functions, Des. Codes Cryptogr., 87 (2019), 2461-2467.  doi: 10.1007/s10623-019-00632-x.

[6]

M. Buratti, J. Yan and C. Wang, From a $1$-rotational RBIBD to a partitioned difference family, Electronic J. Combin., 17 (2010), Research Paper 139, 23pp.

[7]

C. Ding and J. Yin, Combinatorial constructions of optimal constant composition codes, IEEE Trans. Inform. Theory, 51 (2005), 3671-3674.  doi: 10.1109/TIT.2005.855612.

[8]

S. Furino, Difference families from rings, Discret. Math., 97 (1991), 177-190.  doi: 10.1016/0012-365X(91)90433-3.

[9]

D. Jungnickel, Composition theorems for difference families and regular planes, Discrete Math., 23 (1978), 151-158.  doi: 10.1016/0012-365X(78)90113-9.

[10]

D. Jungnickel, On difference matrices and regular Latin squares, Abh. Math. Sem. Univ. Hamburg, 50 (1980), 219-231.  doi: 10.1007/BF02941430.

[11]

D. Jungnickel, On automorphism groups of divisible designs, Canadian J. Math., 34 (1982), 257-297.  doi: 10.4153/CJM-1982-018-x.

[12]

D. Jungnickel, A. Pott and K. W. Smith, Difference sets, in Handbook of Combinatorial Designs, Second Edition, Chapman & Hall/CRC, Boca Raton, FL, (2006), 419–435.

[13]

S. LiH. Wei and G. Ge, Generic constructions for partitioned difference families with applications: A unified combinatorial approach, Des. Codes Cryptogr., 82 (2017), 583-599.  doi: 10.1007/s10623-016-0182-y.

[14]

S. Xu, L. Qu and X. Cao, Three classes of partitioned difference families and their optimal constant composition codes, Adv. Math. Commun., (2020). doi: 10.3934/amc.2020120.

[15]

X. ZengH. CaiX. Tang and Y. Yang, Optimal frequency hopping sequences of odd length, IEEE Trans. Inf. Theory, 59 (2013), 3237-3248.  doi: 10.1109/TIT.2013.2237754.

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