Generalized balanced tournament designs and related codes

A generalized balanced tournament design, or a GBTD(k, m) in short, is a (km, k, k − 1)-BIBD defined on a km-set V. Its blocks can be arranged into an m × (km − 1) array in such a way that (1) every element of V is contained in exactly one cell of each column, and (2) every element of V is contained in at most k cells of each row. In this paper, we present a new construction for GBTDs and show that a GBTD(4, m) exists for any integer m ≥ 5 with at most eight possible exceptions. A link between a GBTD(k, m) and a near constant composition code is also mentioned. The derived code is optimal in the sense of its size.

[1]  Richard M. Wilson,et al.  Constructions and Uses of Pairwise Balanced Designs , 1975 .

[2]  Charles J. Colbourn,et al.  The Existence of Kirkman Squares—Doubly Resolvable (v,3,1)-BIBDs , 2002, Des. Codes Cryptogr..

[3]  E. Lamken.  Generalized balanced tournament designs , 1990 .

[4]  Morris Plotkin,et al.  Binary codes with specified minimum distance , 1960, IRE Trans. Inf. Theory.

[5]  Cunsheng Ding,et al.  Combinatorial constructions of optimal constant-composition codes , 2005, IEEE Transactions on Information Theory.

[6]  Hao Shen,et al.  The PBD-Closure of Constant-Composition Codes , 2007, IEEE Transactions on Information Theory.

[7]  Jie Yan,et al.  Constructions of optimal GDRP(n, lambda;v)'s of type lambda1µm-1 , 2008, Discret. Appl. Math..

[8]  Charles J. Colbourn,et al.  On constant composition codes , 2006, Discret. Appl. Math..

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

[10]  R. Mullin,et al.  Construction of Room Squares , 1968 .

[11]  W. D. Wallis,et al.  On the existence of room squares , 1972 .

[12]  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.

[13]  Douglas R. Stinson,et al.  Frames for Kirkman triple systems , 1987, Discret. Math..

[14]  Esther R. Lamken.  Constructions for generalized balanced tournament designs , 1994, Discret. Math..

[15]  Cunsheng Ding,et al.  A Construction of Optimal Constant Composition Codes , 2006, Des. Codes Cryptogr..

[16]  Charles J. Colbourn,et al.  Pairwise Balanced Designs with Consecutive Block Sizes , 1997, Des. Codes Cryptogr..