Strong difference families, difference covers, and their applications for relative difference families

In (M. Buratti, J Combin Des 7:406–425, 1999), Buratti pointed out the lack of systematic treatments of constructions for relative difference families. The concept of strong difference families was introduced to cover such a problem. However, unfortunately, only a few papers consciously using the useful concept have appeared in the literature in the past 10 years. In this paper, strong difference families, difference covers and their connections with relative difference families and optical orthogonal codes are discussed.

[1]  Anita Pasotti,et al.  Combinatorial designs and the theorem of Weil on multiplicative character sums , 2009, Finite Fields Their Appl..

[2]  Ryoh Fuji-Hara,et al.  Optical orthogonal codes: Their bounds and new optimal constructions , 2000, IEEE Trans. Inf. Theory.

[3]  Fan Chung Graham,et al.  Optical orthogonal codes: Design, analysis, and applications , 1989, IEEE Trans. Inf. Theory.

[4]  Clement W. H. Lam,et al.  Difference Families , 2001, Des. Codes Cryptogr..

[5]  Yanxun Chang,et al.  Combinatorial constructions of optimal optical orthogonal codes with weight 4 , 2003, IEEE Trans. Inf. Theory.

[6]  Siu Lun Ma,et al.  A survey of partial difference sets , 1994, Des. Codes Cryptogr..

[7]  Siu Lun Ma,et al.  New Hadamard matrices of order 4p2 obtained from Jacobi sums of order 16 , 2006, J. Comb. Theory, Ser. A.

[8]  Qing Xiang,et al.  Constructions of Partial Difference Sets and Relative Difference Sets Using Galois Rings II , 1996, J. Comb. Theory, Ser. A.

[9]  ChungF. R.K.,et al.  Optical orthogonal codes , 2006 .

[10]  J. Davis.  Partial difference sets inp-groups , 1994 .

[11]  M. Buratti.  Recursive constructions for difference matrices and relative difference families , 1998 .

[12]  Jianxing Yin,et al.  Some combinatorial constructions for optical orthogonal codes , 1998, Discret. Math..

[13]  Mieko Yamada,et al.  Supplementary difference sets and Jacobi sums , 1992, Discret. Math..

[14]  C. Colbourn,et al.  CORR 99-01 Applications of Combinatorial Designs to Communications , Cryptography , and Networking , 1999 .

[15]  Samuel S. Wagstaff,et al.  Sums of Squares of Integers , 2005 .

[16]  Richard M. Wilson,et al.  Cyclotomy and difference families in elementary abelian groups , 1972 .

[17]  Michael Rosen,et al.  A classical introduction to modern number theory , 1982, Graduate texts in mathematics.

[18]  G. Ge,et al.  Starters and related codes , 2000 .

[19]  T. Storer.  Cyclotomy and difference sets , 1967 .

[20]  Marco Buratti,et al.  Strong difference families over arbitrary graphs , 2008 .

[21]  K. T. Arasu,et al.  Regular Difference Covers , 2005 .

[22]  Ryoh Fuji-Hara,et al.  Optimal (9v, 4, 1) Optical Orthogonal Codes , 2001, SIAM J. Discret. Math..

[23]  Yanxun Chang,et al.  Optimal (4up, 5, 1) optical orthogonal codes , 2004 .

[24]  G. Ge,et al.  Constructions for optimal (v, 4, 1) optical orthogonal codes , 2001, IEEE Trans. Inf. Theory.

[25]  Hanfried Lenz,et al.  Design theory , 1985 .

[26]  Marco Buratti.  Old and new designs via difference multisets and strong difference families , 1999 .

[27]  C. Colbourn,et al.  The CRC handbook of combinatorial designs , edited by Charles J. Colbourn and Jeffrey H. Dinitz. Pp. 784. $89.95. 1996. ISBN 0-8493-8948-8 (CRC). , 1997, The Mathematical Gazette.

[28]  Yanxun Chang,et al.  A New Class of Optimal Optical Orthogonal Codes With Weight Five , 2004, IEEE Trans. Inf. Theory.

[29]  K. T. Arasu,et al.  Cyclic difference covers , 2005, Australas. J Comb..

[30]  John B. Polhill.  New negative Latin square type partial difference sets in nonelementary abelian 2-groups and 3-groups , 2008, Des. Codes Cryptogr..

[31]  Marco Buratti,et al.  Cyclic Designs with Block Size 4 and Related Optimal Optical Orthogonal Codes , 2002, Des. Codes Cryptogr..

[32]  Siu Lun Ma,et al.  Proper partial geometries with Singer groups and pseudogeometric partial difference sets , 2008, J. Comb. Theory, Ser. A.