Abstract
In this paper we discuss the existence problem for a semi-cyclic holey group divisible design of type \((n,m^t)\) with block size \(3\), which is denoted by a \(3\)-SCHGDD of type \((n,m^t)\). When \(n=3\), a \(3\)-SCHGDD of type \((3,m^t)\) is equivalent to a \((3,mt;m)\)-cyclic holey difference matrix, denoted by a \((3,mt;m)\)-CHDM. It is shown that there is a \((3,mt;m)\)-CHDM if and only if \((t-1)m\equiv 0\ (\mathrm{mod}\ 2)\) and \(t\ge 3\) with the exception of \(m\equiv 0\ (\mathrm{mod}\ 2)\) and \(t=3\). When \(n\ge 4\), the case of \(t\) odd is considered. It is established that if \(t\equiv 1\ (\mathrm{mod}\ 2)\) and \(n\ge 4\), then there exists a \(3\)-SCHGDD of type \((n,m^t)\) if and only if \(t\ge 3\) and \((t-1)n(n-1)m\equiv 0\ (\mathrm{mod}\ 6)\) with some possible exceptions of \(n=6\) and \(8\). The main results in this paper have been used to construct optimal two-dimensional optical orthogonal codes with weight \(3\) and different auto- and cross-correlation constraints by the authors recently.
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Acknowledgments
Supported by the Fundamental Research Funds for the Central Universities under Grants \(2011\)JBM\(298,\,2013\)JB\(Z005\) (T. Feng), \(2011\)JBZ\(012\) (Y. Chang), and the NSFC under Grant \(10901016\) (T. Feng), the NSFC under Grants \(11126123\) and \(11201252\) (X. Wang), the NSFC under Grant \(11271042\) (Y. Chang). The authors would like to thank the editor and the two anonymous referees for their helpful comments and valuable suggestions to improve the readability of this paper.
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Communicated by C. J. Colbourn.
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Feng, T., Wang, X. & Chang, Y. Semi-cyclic holey group divisible designs with block size three. Des. Codes Cryptogr. 74, 301–324 (2015). https://doi.org/10.1007/s10623-013-9859-7
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DOI: https://doi.org/10.1007/s10623-013-9859-7
Keywords
- Holey group divisible design
- Semi-cyclic
- Cyclic holey difference matrix
- Cyclic difference family
- Perfect difference family