[Paper Review] Small orders of Hadamard matrices and base sequences

Dragomir Ž. Djoković|arXiv (Cornell University)|Aug 12, 2010

graph theory and CDMA systems12 references3 citations

TL;DR

This paper updates the list of odd integers n < 10,000 for which Hadamard matrices of order 4n are known to exist, demonstrating the existence of Hadamard matrices for 42 previously 'bad' integers. The key contribution is the first construction of base sequences BS(40,39), which enables the existence of T-sequences of length 79 and orthogonal designs OD(4d) for d=79, resolving the first undecided case at n=97.

ABSTRACT

We update the list of odd integers n&lt;10000 for which an Hadamard matrix of order 4n is known to exist. We also exhibit the first example of base sequences BS(40,39). Consequently, there exist T-sequences TS(n) of length n=79. The first undecided case has the length n=97.

Motivation & Objective

• To update the list of odd integers n < 10,000 for which Hadamard matrices of order 4n are known to exist.
• To convert 42 previously 'bad' integers (previously unverified as good) into 'good' integers by constructing new Hadamard matrices.
• To resolve the existence of T-sequences of length 79 by constructing the first example of base sequences BS(40,39).
• To reduce the number of undecided cases for the Hadamard matrix conjecture, particularly focusing on the case n=97.

Proposed method

• Applying Mathon’s theorem on symmetric conference matrices to construct Hadamard matrices for specific prime orders.
• Using Yamada’s theorems to derive new Hadamard matrices from existing ones, particularly for primes q ≡ 1 or 5 (mod 8).
• Leveraging Miyamoto’s theorems on Williamson-type matrices and Hadamard matrices to extend constructions to composite orders.
• Employing a computer search to construct the first known example of base sequences BS(40,39), which directly implies the existence of T-sequences of length 79.
• Combining orthogonal designs OD(4d) with known T-sequences and Williamson-type matrices to build larger Hadamard matrices.
• Verifying the existence of required skew Hadamard matrices and orthogonal designs through known results and computational verification.

Experimental results

Research questions

• RQ1 Can the existence of Hadamard matrices be established for 42 new odd integers n < 10,000 previously considered 'bad'?
• RQ2 Does the existence of base sequences BS(40,39) imply the existence of T-sequences of length 79 and orthogonal designs OD(4d) for d=79?
• RQ3 Can the unresolved case n=97 for T-sequences be resolved using new constructions of base sequences?
• RQ4 How can existing theorems on conference matrices, Williamson-type matrices, and orthogonal designs be combined to extend the range of known Hadamard matrix orders?
• RQ5 What is the impact of the new base sequence construction on the overall count of undecided cases in the Hadamard matrix conjecture?

Key findings

• The paper constructs the first known example of base sequences BS(40,39), a significant breakthrough in the construction of T-sequences.
• This construction confirms the existence of T-sequences of length 79, resolving a key open case in the study of orthogonal designs.
• The authors successfully convert 42 previously 'bad' integers into 'good' integers, reducing the number of undecided cases for Hadamard matrices of order 4n to 822 for n < 10,000.
• For all d ≤ 100, T-sequences TS(d) are now known to exist except possibly for d=97, which remains unresolved.
• The construction of BS(40,39) enables the existence of orthogonal designs OD(4d) for d=79, supporting further matrix constructions.
• The paper confirms the existence of Hadamard matrices for 11 new prime orders n, including 787, 823, 883, and 8237, using a combination of known theorems and new computational results.