[Paper Review] Embedding cocyclic D-optimal designs in cocyclic Hadamard matrices

Víctor Álvarez, José-Andrés Armario+2 more|arXiv (Cornell University)|Jan 20, 2012

graph theory and CDMA systems13 references3 citations

TL;DR

This paper presents a method to embed cocyclic D-optimal designs—submatrices with maximal determinant—into cocyclic Hadamard matrices of order $4t$, leveraging algebraic cocycle operations. The key result shows that for $t=5$, a D-optimal design of order 10 can be embedded in a CP (completely pivoted) form within any $D_{20}$-Hadamard matrix, yielding a specific pivot pattern and confirming that such submatrices attain the largest possible minors at each step of Gaussian elimination with complete pivoting.

ABSTRACT

In this paper a method for embedding cocyclic submatrices with ``large'' determinants of orders 2t in certain cocyclic Hadamard matrices of orders 4t is described (t an odd integer). If these determinants attain the largest possible value, we are embedding D-optimal designs. Applications to the pivot values that appear when Gaussian Elimination with complete pivoting is performed on these cocyclic Hadamard matrices are studied.

Motivation & Objective

• To develop a method for embedding cocyclic submatrices with large determinants into cocyclic Hadamard matrices of order $4t$.
• To determine under what conditions such embedded submatrices achieve the maximal determinant, i.e., become D-optimal designs.
• To analyze the pivot values arising during Gaussian elimination with complete pivoting on these matrices.
• To establish a link between the structure of cocyclic Hadamard matrices and the feasibility of achieving maximal minors at each step of the elimination process.

Proposed method

• The method uses combinatorial operations—row and column elimination and addition—on cocyclic matrices to embed submatrices with large determinants.
• These operations are translated into an algebraic framework using cocycles, enabling systematic construction of submatrices within the cocyclic structure.
• The approach relies on verifying that the determinant of the embedded $2t \times 2t$ submatrix matches the theoretical upper bound for $(-1,1)$-matrices.
• The feasibility of the embedded submatrix is confirmed by checking that its $k\times k$ minors are maximal among all $k\times k$ minors fixing the first $k-1$ rows and columns.
• The proof uses known results on the spectrum of determinants for $(-1,1)$-matrices to bound intermediate minors and confirm maximality.
• The method is applied to $t=5$, embedding a $10\times 10$ D-optimal design into a $20\times 20$ cocyclic Hadamard matrix, and verifying its CP property.

Experimental results

Research questions

• RQ1 Can a $2t \times 2t$ D-optimal design be embedded in a cocyclic Hadamard matrix of order $4t$ using algebraic cocycle operations?
• RQ2 Under what conditions does a submatrix with maximal determinant appear in the top-left corner of a CP matrix equivalent to a given Hadamard matrix?
• RQ3 What pivot pattern emerges during Gaussian elimination with complete pivoting when the matrix contains a D-optimal design?
• RQ4 Is the existence of a submatrix with maximal determinant sufficient to guarantee that it appears as the leading minor in some CP equivalent matrix?
• RQ5 Can the determinant of a submatrix be used to infer the pivot structure in Gaussian elimination with complete pivoting?

Key findings

• For $t=5$, a $10\times 10$ D-optimal design with determinant $125 \cdot 2^9 = 64000$ can be embedded in a $20\times 20$ cocyclic Hadamard matrix.
• The embedded $D_{10}$ matrix achieves the maximum possible $k\times k$ minor for all $k \leq 10$, confirming its feasibility as a CP matrix.
• The pivot pattern $(1,2,2,4,3,10/3,16/5,5,24/5,6)$ for the first ten pivots is confirmed to appear in all three equivalence classes of $D_{20}$-Hadamard matrices.
• The study confirms that the existence of a submatrix with maximal determinant implies that it can be brought to the top-left corner in a CP equivalent matrix, at least for the $D_{10}$ case.
• The method successfully embeds the $D_{10}$ design into a $D_{20}$-Hadamard matrix constructed via composition of specific cocyclic matrices $M_{\partial_2}, M_{\partial_4}, \dots, M_{\gamma}$.
• The determinant of the resulting $20\times 20$ matrix $M_\psi$ satisfies $\det(M_\psi) = 125 \cdot 2^9$, confirming the submatrix's optimality.