Constructions of augmented orthogonal arrays

Augmented orthogonal arrays (AOAs) were introduced by Stinson, who showed the equivalence between ideal ramp schemes and augmented orthogonal arrays (Discrete Math. 341 (2018), 299-307). In this paper, we show that there is an AOA$(s,t,k,v)$ if and only if there is an OA$(t,k,v)$ which can be partitioned into $v^{t-s}$ subarrays, each being an OA$(s,k,v)$, and that there is a linear AOA$(s,t,k,q)$ if and only if there is a linear maximum distance separable (MDS) code of length $k$ and dimension $t$ over $\mathbb{F}_q$ which contains a linear MDS subcode of length $k$ and dimension $s$ over $\mathbb{F}_q$. Some constructions for AOAs and some new infinite classes of AOAs are also given.

[1]  Svenja Huntemann,et al.  The upper bound of general Maximum Distance Separable codes , 2012 .

[2]  Charles C. Lindner,et al.  Steiner Quadruple Systems , 2008 .

[3]  Gennian Ge,et al.  On (g, 4;1)-difference matrices , 2005, Discret. Math..

[4]  Hedvig Mohácsy.  The Asymptotic Existence of Group Divisible t ‐Designs of Large Order with Index One , 2013 .

[5]  Lijun Ji,et al.  Constructions of new orthogonal arrays and covering arrays of strength three , 2010, J. Comb. Theory, Ser. A.

[6]  Jennifer Werfel,et al.  Orthogonal Arrays Theory And Applications , 2016 .

[7]  K. A. Bush.  A Generalization of a Theorem due to MacNeish , 1952 .

[8]  K. A. Bush.  Orthogonal Arrays of Index Unity , 1952 .

[9]  Douglas R. Stinson,et al.  Ideal ramp schemes and related combinatorial objects , 2018, Discret. Math..

[10]  Simeon Ball,et al.  On sets of vectors of a finite vector space in which every subset of basis size is a basis II , 2012, Designs, Codes and Cryptography.

[11]  Solomon W. Golomb,et al.  A new recursive construction for optical orthogonal codes , 2003, IEEE Trans. Inf. Theory.

[12]  Adi Shamir,et al.  How to share a secret , 1979, CACM.

[13]  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.

[14]  Yang Li,et al.  On the existence of orthogonal arrays OA(3, 5, 4n+2) , 2011, J. Comb. Theory, Ser. A.

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

[16]  Ron M. Roth,et al.  Introduction to Coding Theory , 2019, Discrete Mathematics.

[17]  Catherine A. Meadows,et al.  Security of Ramp Schemes , 1985, CRYPTO.