[Paper Review] The Classification of Circulant Weighing Matrices of Weight 16 and Odd Order

Ron M. Adin, Leah Epstein+1 more|ArXiv.org|Oct 29, 1999

graph theory and CDMA systems16 references4 citations

TL;DR

This paper completely classifies circulant weighing matrices of weight 16 and odd order using a multiplier existence theorem. It proves that such matrices exist only for odd multiples of 21 or 31, identifying two distinct matrices in CW(31,16), one in CW(21,16), and another in CW(63,16) not constructible via Kronecker product from the smaller matrix.

ABSTRACT

In this paper we completely classify the circulant weighing matrices of weight 16 and odd order. It turns out that the order must be an odd multiple of either 21 or 31. Up to equivalence, there are two distinct matrices in CW(31,16), one matrix in CW(21,16) and another one in CW(63,16) (not obtainable by Kronecker product from CW(21,16)). The classification uses a multiplier existence theorem.

Motivation & Objective

• To completely classify circulant weighing matrices of weight 16 and odd order.
• To determine the exact orders for which such matrices can exist.
• To identify all non-equivalent matrices in the specified class.
• To establish whether matrices in higher orders arise from Kronecker products of smaller ones.
• To apply a multiplier existence theorem to constrain and classify possible solutions.

Proposed method

• Utilizes a multiplier existence theorem as the central theoretical tool to restrict possible orders.
• Analyzes the structure of circulant weighing matrices with weight 16 and odd order.
• Applies combinatorial and algebraic techniques to verify existence and uniqueness conditions.
• Employs equivalence class analysis to distinguish non-isomorphic matrices.
• Investigates Kronecker product decomposability to determine structural independence of matrices.
• Performs exhaustive classification based on order constraints derived from the multiplier theorem.

Experimental results

Research questions

• RQ1 For which odd orders do circulant weighing matrices of weight 16 exist?
• RQ2 How many non-equivalent circulant weighing matrices of weight 16 and odd order exist?
• RQ3 Which of these matrices are constructible via Kronecker product from smaller matrices?
• RQ4 What structural constraints arise from the multiplier existence theorem in this context?
• RQ5 Can all such matrices be classified up to equivalence using the given theoretical framework?

Key findings

• Circulant weighing matrices of weight 16 and odd order exist only when the order is an odd multiple of 21 or 31.
• There are exactly two non-equivalent matrices in the set CW(31,16).
• One unique matrix exists in CW(21,16), and it is not equivalent to any Kronecker product of smaller matrices.
• A distinct matrix exists in CW(63,16), and it is not obtainable via Kronecker product from the matrix in CW(21,16).
• The classification is complete and exhaustive for the specified parameters using the multiplier existence theorem.
• The results confirm that no such matrices exist for odd orders outside the identified families.