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Generalizing the Bierbrauer-Friedman bound for orthogonal arrays

Denis S. Krotov, Ferruh Özbudak, Vladimir N. Potapov

TL;DR

This work generalizes the Bierbrauer--Friedman bound to mixed-level orthogonal arrays by recasting OA as algebraic $t$-designs on a mixed Hamming graph and showing bound-attaining arrays correspond to radius-1 completely regular codes. It further develops an additive, prime-power level framework via multispreads that characterizes bound-attaining mixed-level OAs as CR codes and multispread partitions, accompanied by explicit constructions. A polynomial generalization based on distance-regular graphs yields BF-type bounds beyond the traditional regime, delivering new lower bounds for pure-level arrays in cases where the original bound is trivial. Collectively, the results unify OA theory, coding theory, and spectral graph methods for mixed-level designs and open avenues for non-prime-power constructions and further bounds.

Abstract

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 radius-1 completely regular codes (equivalently, intriguing sets, equitable 2-partitions, perfect 2-colorings) in the corresponding multigraph. For the case when the numbers of levels are powers of the same prime number, we characterize, in terms of multispreads, additive mixed-level orthogonal arrays attaining the BF bound. For pure-level orthogonal arrays, we consider versions of the BF bound obtained by replacing the Hamming graph by its polynomial generalization and show that in some cases this gives a new bound. Keywords: orthogonal array, algebraic t-design, completely regular code, equitable partition, intriguing set, Hamming graph, Bierbrauer-Friedman bound, additive codes.

Generalizing the Bierbrauer-Friedman bound for orthogonal arrays

TL;DR

This work generalizes the Bierbrauer--Friedman bound to mixed-level orthogonal arrays by recasting OA as algebraic -designs on a mixed Hamming graph and showing bound-attaining arrays correspond to radius-1 completely regular codes. It further develops an additive, prime-power level framework via multispreads that characterizes bound-attaining mixed-level OAs as CR codes and multispread partitions, accompanied by explicit constructions. A polynomial generalization based on distance-regular graphs yields BF-type bounds beyond the traditional regime, delivering new lower bounds for pure-level arrays in cases where the original bound is trivial. Collectively, the results unify OA theory, coding theory, and spectral graph methods for mixed-level designs and open avenues for non-prime-power constructions and further bounds.

Abstract

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 radius-1 completely regular codes (equivalently, intriguing sets, equitable 2-partitions, perfect 2-colorings) in the corresponding multigraph. For the case when the numbers of levels are powers of the same prime number, we characterize, in terms of multispreads, additive mixed-level orthogonal arrays attaining the BF bound. For pure-level orthogonal arrays, we consider versions of the BF bound obtained by replacing the Hamming graph by its polynomial generalization and show that in some cases this gives a new bound. Keywords: orthogonal array, algebraic t-design, completely regular code, equitable partition, intriguing set, Hamming graph, Bierbrauer-Friedman bound, additive codes.

Paper Structure

This paper contains 6 sections, 9 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Definitions and notations
  3. Orthogonal arrays and algebraic designs
  4. Additive mixed-level orthogonal arrays attaining the Bierbrauer--Friedman bound
  5. Polynomial generalization of the Bierbrauer--Friedman bound
  6. Conclusion

Key Result

Lemma 6

The following functions form an orthogonal basis from eigenfunctions of $H(q_1\cdot q_2 \cdot \ldots \cdot q_n)$: where $\xi_i$ is the degree-$q_i$ primitive root of $1$. Moreover, $\chi_{(b_1,b_2,...,b_n)}$ is an eigenfunction corresponding to the eigenvalue $\theta_w = k - w Q$, where is the degree of $H(q_1{\cdot}q_2{\cdot}...{\cdot}q_n)$, $Q=\mathrm{lcm}(q_1,...,q_n)$, and $w$ is the number

Theorems & Definitions (29)

  • Definition 1: adjacency matrix of a graph, eigenfunctions and eigenspaces
  • Definition 2: completely regular code with covering radius $1$, $\{b;c\}$-CR code
  • Remark 1
  • Definition 3: $H(q_1\cdot q_2\cdot\ldots\cdot q_n)$, Hamming graph
  • Definition 4: algebraic $t$-design
  • Definition 5: orthogonal arrays, OA$(N,q_1{\cdot} q_2{\cdot} \ldots{\cdot} q_n , t)$
  • Lemma 6
  • proof
  • proof
  • Lemma 8: KroPot:CRCEP
  • ...and 19 more