ISSN 0032-9460, Problems of Information Transmission, 2013, Vol. 49, No. 4, pp. 322–332.
Pleiades Publishing, Inc., 2013.
Original Russian Text
P. Boyvalenkov, H. Kulina, 2013, published in Problemy Peredachi Informatsii, 2013, Vol. 49, No. 4, pp. 28–40.
Investigation of Binary Orthogonal Arrays
via Their Distance Distributions
and H. Kulina
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, Bulgaria
Received November 13, 2012; in ﬁnal form, August 14, 2013
Dedicated to Stefan Dodunekov
(September 5, 1945 – August 5, 2012)
Abstract—We show how one can use polynomial techniques to compute all possible distance
distributions of binary orthogonal arrays (OAs) of relatively small lengths and strengths. Then
we exploit certain connections between OAs and their derived OAs. Having all distance distri-
butions of OAs under consideration, we are able to test them aimed at classiﬁcation results.
Let H(n, 2) be the binary Hamming space of dimension n. An orthogonal array C of strength τ
and index λ,orequivalentlyaτ-design, in H(n, 2) consists of rows of an M × n matrix with the
property that every M × τ submatrix contains all ordered τ -tuples of H(τ,2) as rows, each one
exactly λ =
times. Note that this deﬁnition allows repetition of rows. We denote C by τ-(n, M).
The book  presents a nice study on OAs (not only binary) with descriptions of their connections
to statistics and coding theory and with applications to computer science and cryptography. This
book features all of the key results, many very useful tables, and a large number of examples,
exercises, and research problems. In particular, the authors say that “ﬁnding the smallest possible
number of rows is a problem of eminent importance.”
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 [1, Table 12.1]. In this
paper we present method for investigation of OAs by their distance distributions. This method can
be used in searches for values of the function L(n, τ).
In H(n, 2) with the Hamming distance d(x, y), x, y ∈ H(n, 2), and inner product x, y =
, we have the following equivalent deﬁnition of OAs (cf. ), which is convenient for the
so-called polynomial techniques.
Supported by the Bulgarian NSF, contract no. I01/0003.
Supported in part by the NPD, Plovdiv University, project no. NI13 FMI02.