ISSN 0032-9460, Problems of Information Transmission, 2015, Vol. 51, No. 4, pp. 326–334.
Pleiades Publishing, Inc., 2015.
Original Russian Text
P. Boyvalenkov, H. Kulina, T. Marinova, M. Stoyanova, 2015, published in Problemy Peredachi Informatsii, 2015, Vol. 51,
No. 4, pp. 23–31.
Nonexistence of Binary Orthogonal Arrays
via Their Distance Distributions
, H. Kulina
, and M. Stoyanova
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Soﬁa, Bulgaria
Faculty of Mathematics and Natural Sciences, South-Western University, Blagoevgrad, Bulgaria
Faculty of Mathematics and Informatics, Plovdiv University, Plovdiv, Bulgaria
Faculty of Mathematics and Informatics, Soﬁa University, Soﬁa, Bulgaria
e-mail: email@example.com, firstname.lastname@example.org
Received December 20, 2014; in ﬁnal form, July 20, 2015
Abstract—We investigate binary orthogonal arrays by making use of the fact that all possible
distance distributions of the arrays under investigation and of related arrays can be computed.
We apply certain relations for reducing the number of feasible distance distributions. In some
cases this leads to nonexistence results. In particular, we prove that there exist no binary
orthogonal arrays with parameters (strength, length, cardinality) = (4, 10, 6 · 2
), (4, 11, 6 · 2
(4, 12, 7 · 2
), (5, 11, 6 · 2
), (5, 12, 6 · 2
), and (5, 13, 7 · 2
Let H(n, 2) be the binary Hamming space of dimension n. An orthogonal array (OA) of
strength τ and index λ in H(n, 2) (or a binary orthogonal array, BOA) consists of the rows of
an M × n matrix C with the property that every M × τ submatrix of C contains all ordered
τ-tuples of H(τ,2) as rows, each one exactly λ = M/2
times. Note that this deﬁnition allows
repetition of rows. We denote C by (τ,n,M).
In  a method for investigating BOAs via calculation and subsequent analysis of their distance
distributions was proposed. In this paper we add further arguments by using relations between
(τ,n,M = λ2
)and(τ − 1,n− 1,M
) BOAs. This makes it possible to eliminate more
possibilities and, combined with other information from the method of , gives certain nonexistence
We prove that there exist no BOAs with parameters (strength, length, cardinality) = (τ,n =
τ +6,M =6· 2
), (τ, n = τ +7,M =6· 2
), and (τ,n = τ +8,M =7· 2
)forτ ≥ 4. Other our
nonexistence results (for example (τ,n = τ +4,M =6· 2
)and(τ,n = τ +4,M =7· 2
)forτ ≥ 8)
follow also by a result of  (see Section 4.3).
Denote by L(n, τ ) the minimum possible index λ such that a binary (τ,n, M = λ2
) OA exists.
A table with all possible values of L(n, τ)forn ≤ 32 and τ ≤ 10 is given in [3, Table 12.1]. Our
nonexistence results show that L(10, 4) = L(11, 5) ≥ 7, L(11, 4) = L(12, 5) ≥ 7, and L(12, 4) =
L(13, 5) = 8. This improves the corresponding ﬁve entries in Table 12.1 from .
Supported by the Bulgarian National Science Foundation under Contract I01/0003.
Supported in part by the NPD, Plovdiv University, Bulgaria, project NI15 FMI-004.
Supported in part by the Science Foundation of Soﬁa University, Bulgaria, under Contract 015/2014.