Related papers: On the OA(1536,13,2,7) and related orthogonal arra…

We describe the classification of orthogonal arrays OA$(2048,14,2,7)$, or, equivalently, completely regular $\{14;2\}$-codes in the $14$-cube ($30848$ equivalence classes). In particular, we find that there is exactly one…

We describe the computer-aided classification of equitable partitions of the $12$-cube with quotient matrix $[[2,10],[6,6]]$, or, equivalently, simple orthogonal arrays OA$(1536,12,2,7)$, or order-$7$ correlation-immune Boolean functions in…

An orthogonal array (OA), denoted by $\text{OA}(M, n, q, t)$, is an $M \times n$ matrix over an alphabet of size $q$ such that every selection of $t$ columns contains each possible $t$-tuple exactly $\lambda=M / q^t$ times. An irredundant…

We study properties of binary codes with parameters close to the parameters of 1-perfect codes. An arbitrary binary $(n=2^m-3, 2^{n-m-1}, 4)$ code $C$, i.e., a code with parameters of a triply-shortened extended Hamming code, is a cell of…

We characterize mixed-level orthogonal arrays in terms of algebraic designs in a special multigraph. We prove a mixed-level analog of the Bierbrauer-Friedman (BF) bound for pure-level orthogonal arrays and show that arrays attaining it are…

Mixed (asymmetric) orthogonal arrays (MOAs) generalize classical orthogonal arrays by allowing columns over different alphabets. However, their study requires very different structural tools than those used for symmetric orthogonal arrays…

We show that any binary $(n=2^m-3, 2^{n-m}, 3)$ code $C_1$ is a part of an equitable partition (perfect coloring) $\{C_1,C_2,C_3,C_4\}$ of the $n$-cube with the parameters $((0,1,n-1,0)(1,0,n-1,0)(1,1,n-4,2)(0,0,n-1,1))$. Now the…

We consider the problem of existence of perfect $2$-colorings (equitable $2$-partitions) of Hamming graphs with given parameters. We start with conditions on parameters of graphs and colorings that are necessary for their existence. Next we…

An orthogonal array OA(q^{2n-1},q^{2n-2}, q,2) is constructed from the action of a subset of PGL(n+1,q^2) on some non--degenerate Hermitian varieties in PG(n,q^2). It is also shown that the rows of this orthogonal array correspond to some…

We generate all the Orthogonal Arrays (OAs) of a given size n and strength t as the union of a collection of OAs which belong to an inclusion-minimal set of OAs. We derive a formula for computing the (Generalized) Word Length Pattern of a…

Classifying orthogonal arrays is a well known important class of problems that asks for finding all non-isomorphic, non-negative integer solutions to a class of systems of constraints. Solved instances are scarce. We develop two new methods…

We construct orthogonal arrays OA$_{\lambda} (k,n)$ (of strength two) having a row that is repeated $m$ times, where $m$ is as large as possible. In particular, we consider OAs where the ratio $m / \lambda$ is as large as possible; these…

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…

An Orthogonally resolvable Matching Design OMD$(n, k)$ is a partition of the edges the complete graph $K_n$ into matchings of size $k$, called blocks, such that the blocks can be resolved in two different ways. Such a design can be…

We introduce a construction and an algorithm, both based on Topological Data Analysis (TDA), to tackle the problem of the isomorphism check of Orthogonal Arrays (OAs). Specifically, we associate to any binary OA a persistence diagram, one…

Orthogonal arrays are a type of combinatorial design that were developed in the 1940s in the design of statistical experiments. In 1947, Rao proved a lower bound on the size of any orthogonal array, and raised the problem of constructing…

We develop and apply combinatorial algorithms for investigation of the feasible distance distributions of binary orthogonal arrays with respect to a point of the ambient binary Hamming space utilizing constraints imposed from the relations…

We give a method of constructing self-orthogonal codes from equitable partitions of association schemes. By applying this method we construct self-orthogonal codes from some distance-regular graphs. Some of the obtained codes are optimal.…

It is known that the order of correlation immunity of a nonconstant unbalanced Boolean function in $n$ variables cannot exceed $2n/3-1$; moreover, it is $2n/3-1$ if and only if the function corresponds to an equitable $2$-partition of the…

Let $(\Omega,\mathscr F,\mathbb P) $ be a probability space and let $(\mathscr F_n)$ be a binary filtration, i.e. exactly one atom of $\mathscr F_{n-1}$ is divided into two atoms of $\mathscr F_n$ without any restriction on their respective…