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The asymptotic existence of DR\((v,k,k-1)\)-BIBDs

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Abstract

A Kirkman square with index \(\lambda \), latinicity \(\mu \), block size \(k\), and \(v\) points, \(KS_k(v;\mu ,\lambda )\), is a \(t\times t\) array (\(t=\lambda (v-1)/\mu (k-1)\) ) defined on a \(v\)-set \(V\) such that (1) every point of \(V\) is contained in precisely \(\mu \) cells of each row and column, (2) each cell of the array is either empty or contains a \(k\)-subset of \(V\), and (3) the collection of blocks obtained from the non-empty cells of the array is a \((v,k,\lambda )\)-BIBD. For \(\mu = 1\), the existence of a \(KS_k(v;\mu ,\lambda )\) is equivalent to the existence of a doubly resolvable \((v,k,\lambda )\)-BIBD. The asymptotic existence of \(KS_k(v;1,1)\) was established in 2009. In this paper we establish necessary and sufficient conditions for the asymptotic existence of \(KS_k(v;1,k-1)\) or DR\((v,k,k-1)\)-BIBDs.

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    E. R. Lamken

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Correspondence to E. R. Lamken.

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Dedicated to the memory of Scott Vanstone.

This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Cryptography, Codes, Designs and Finite Fields: In Memory of Scott A. Vanstone”.

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Lamken, E.R. The asymptotic existence of DR\((v,k,k-1)\)-BIBDs. Des. Codes Cryptogr. 77, 553–562 (2015). https://doi.org/10.1007/s10623-015-0090-6

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  • DOI: https://doi.org/10.1007/s10623-015-0090-6

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