[Paper Review] Characterizations of regularity for certain $Q$-polynomial association schemes

Sho Suda|ArXiv.org|Oct 24, 2009

Coding theory and cryptography8 references3 citations

TL;DR

This paper characterizes triple and quadruple regularity in $Q$-polynomial association schemes arising from linked systems of symmetric designs and real mutually unbiased bases (MUB). It proves that triple regularity is equivalent to $a_1^*=0$, and shows that maximal MUB yield quadruply regular schemes, with regularity fully characterized by the parameter $d/2+1$ for dimension $d$. The results establish combinatorial equivalences between extremal configurations and high-order regularity in association schemes.

ABSTRACT

It is shown that linked systems of symmetric designs with $a_1^*=0$ and mutually unbiased bases (MUB) are triply regular association schemes. In this paper, we characterize triple regularity of linked systems of symmetric designs by its Krein number. And we prove that maximal MUB carries a quadruply regular association scheme and characterize the quadruple regularity of MUB by its parameter.

Motivation & Objective

• To characterize triple regularity in $Q$-polynomial association schemes derived from linked systems of symmetric designs.
• To establish a necessary and sufficient condition for quadruple regularity in maximal real mutually unbiased bases (MUB).
• To clarify the combinatorial structure of $4$-class $Q$-polynomial schemes that are both $Q$-bipartite and $Q$-antipodal.
• To investigate whether linked systems of symmetric designs with $a_1^*=0$ can achieve quadruple regularity.
• To determine whether maximal MUB induce quintuply regular association schemes.

Proposed method

• Uses Krein number analysis to characterize triple regularity in linked systems of symmetric designs.
• Applies the Cauchy-Schwarz inequality to count 4-tuples of points and derive necessary conditions for quadruple regularity.
• Employs combinatorial counting via two-way counting of incidence structures involving $X_1$ and $\bigcup_{i=2}^f X_i$ to derive constraints on parameters.
• Analyzes the weight enumerator of a Kerdock-like binary code derived from maximal MUB to show strength-5 orthogonal array properties.
• Leverages known correspondences between maximal MUB and $4$-class $Q$-polynomial schemes that are both $Q$-bipartite and $Q$-antipodal.
• Uses algebraic number theory to show that the equation $v(v-1)(v+3)$ must be a square for quadruple regularity, and proves no solution exists for $v \geq 15$.

Experimental results

Research questions

• RQ1 Is triple regularity in a $3$-class $Q$-polynomial association scheme with $Q$-antipodal property equivalent to $a_1^*=0$?
• RQ2 Can a linked system of symmetric designs with $a_1^*=0$ yield a quadruply regular association scheme?
• RQ3 What is the necessary and sufficient condition for a maximal MUB to induce a quadruply regular association scheme?
• RQ4 Does the existence of a quadruply regular scheme from maximal MUB imply quintuple regularity?
• RQ5 What constraints arise from requiring that the number of common neighbors of four points in a linked system of symmetric designs is constant?

Key findings

• Triple regularity in a $3$-class $Q$-polynomial association scheme with $Q$-antipodal property holds if and only if $a_1^*=0$, confirming the converse of a prior result.
• The equality case in Noda's inequality for linked systems of symmetric designs implies $a_1^*=0$, which is equivalent to triple regularity.
• No linked system of symmetric designs with $1<k<v-1$ and $v\geq15$ can be quadruply regular, as the required equation $v(v-1)(v+3)$ being a square has no solution in the required range.
• Maximal MUB in $\mathbb{R}^d$ with $d>4$ yield a quadruply regular $4$-class association scheme, and the regularity is fully characterized by the parameter $d/2+1$.
• The Kerdock-like code derived from maximal MUB has strength 5, which implies that the number of common neighbors of four points in certain relations is uniquely determined.
• Integralities of intersection numbers in the quadruply regular scheme derived from maximal MUB do not yield new necessary conditions beyond the known $d = k^2/16$.