Further progress on difference families with block size 4 or 5

A strong indication about the existence of a (7p, 4, 1) difference family with p ≡ 7 (mod 12) a prime has been given in [11]. Here, developing some ideas of that paper, we give, much more generally, a strong indication about the existence of a cyclic (pq, 4, 1) difference family whenever p and q are primes congruent to 7 (mod 12) and of a cyclic (pq, 5, 1) difference family whenever p and q are primes congruent to 11 (mod 20). Indeed we give an algorithm for their construction that seems to be always successful and we have checked it works whenever both primes p and q do not exceed 1,000. All our (pq, 4, 1) and (pq, 5, 1) difference families have the nice property of admitting a multiplier of order 3 or 5, respectively, that fixes almost all base blocks. As an intermediate result we also find an optimal (p, 5, 1) optical orthogonal code for every prime p ≡ 11 (mod 20) not exceeding 10,000.

[1]  M. Buratti.  Improving two theorems of bose on difference families , 1995 .

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

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

[4]  R. Julian R. Abel,et al.  Some progress on (v, 4, 1) difference families and optical orthogonal codes , 2004, J. Comb. Theory, Ser. A.

[5]  Rudolf Mathon,et al.  Constructions for Cyclic Steiner 2-designs , 1987 .

[6]  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.

[7]  J. W. P. Hirschfeld,et al.  Some cyclic and 1-rotational designs , 2001 .

[8]  Marco Buratti.  From a (G, k, 1) to a (Ck ⊕ G, k, 1) Difference Family , 1997, Des. Codes Cryptogr..

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

[10]  C. Colbourn,et al.  Recursive constructions for cyclic block designs , 1984 .

[11]  Douglas R Stinson,et al.  Surveys in Combinatorics, 1999: Applications of Combinatorial Designs to Communications, Cryptography, and Networking , 1999 .

[12]  M. Buratti.  On simple radical difference families , 1995 .

[13]  I. Anderson.  Some 2‐rotational and cyclic designs , 1996 .

[14]  Rose Peltesohn.  Eine Lösung der beiden Heffterschen Differenzenprobleme , 1939 .

[15]  K. Chen,et al.  Existence of (q, k, 1) difference families with q a prime power and k = 4, 5 , 1999 .

[16]  R. Julian R. Abel,et al.  Directed-ordered whist tournaments and (v, 5, 1) difference families: existence results and some new classes of Z-cyclic solutions , 2004, Discret. Appl. Math..

[17]  B. Bollobás.  Surveys in Combinatorics , 1979 .

[18]  Marco Buratti,et al.  Perfect Cayley Designs as Generalizations of Perfect Mendelsohn Designs , 2001, Des. Codes Cryptogr..

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

[20]  R. C. Bose.  ON THE CONSTRUCTION OF BALANCED INCOMPLETE BLOCK DESIGNS , 1939 .

[21]  Marco Buratti.  A packing problem its application to Bose's families , 1996 .