An orthogonal array (OA), denoted by OA ( M , n , q , t ) , is an M × n matrix over an alphabet of size q such that every selection of t columns contains each possible t -tuple exactly λ = M / q t times. An irredundant orthogonal array (IrOA) is an OA with the additional property that, in any selection of n − t columns, all resulting rows are distinct. IrOAs were first introduced by Goyeneche and Życzkowski in 2014 to construct t -uniform quantum states without redundant information. Beyond their quantum applications, we focus on IrOAs as a combinatorial and coding theory problem. An OA is an IrOA if and only if its minimum Hamming distance is at least t + 1 . Using this characterization, we demonstrate that for any linear code, either the code itself or its Euclidean dual forms a linear IrOA, giving a huge source of IrOAs. In particular, the self-dual codes yield IrOAs. Moreover, we construct new families of linear IrOAs based on self-dual, Maximum Distance Separable (MDS), and MDS-self-dual codes. Finally, we establish bounds on the minimum distance and covering radius of IrOAs.

We employ signed measures that are positive definite up to certain degrees to establish Levenshtein-type upper bounds on the cardinality of codes with given minimum and maximum distances, and universal lower bounds on the potential energy (for absolutely monotone interactions) for codes with given maximum distance and cardinality. The distance distributions of codes that attain the bounds are found in terms of the parameters of Levenshtein-type quadrature formulas. Necessary and sufficient conditions for the optimality of our bounds are derived. Further, we obtain upper bounds on the energy of codes of fixed minimum and maximum distances and cardinality.

We employ signed measures that are positive definite up to certain degrees to establish Levenshtein-type upper bounds on the cardinality of codes with given minimum and maximum distance, and universal lower bounds on the potential energy (for absolutely monotone interactions) for codes with given maximum distance and fixed cardinality. In particular, we extend the framework of Levenshtein bounds for such codes.

We use linear programming techniques to obtain new upper bounds on the maximal squared minimum distance of spherical codes with fixed cardinality. Functions Q/sub j/(n,s) are introduced with the property that Q/sub j/(n,s)m if and only if the Levenshtein bound L/sub m/(n,s) on A(n,s)=max{|W|:W is an (n,|W|,s) code} can be improved by a polynomial of degree at least m+1. General conditions on the existence of new bounds are presented. We prove that for fixed dimension n/spl ges/5 there exists a constant k=k(n) such that all Levenshtein bounds L/sub m/(n, s) for m/spl ges/2k-1 can be improved. An algorithm for obtaining new bounds is proposed and discussed.