[Paper Review] Superregular matrices over small finite fields

Paulo Almeida, Diego Napp|arXiv (Cornell University)|Aug 1, 2020

Coding theory and cryptography15 references4 citations

TL;DR

This paper investigates lower triangular Toeplitz superregular matrices over small finite prime fields, proposing constructions that minimize the number of distinct nontrivial minors. Using computational search and greedy algorithms in Maple, the authors construct such matrices for orders up to 10 over small prime fields, demonstrating existence for γ = 8, 9, and 10 over fields like F_173 and F_257, and conjecturing bounds on field size and minor counts for general γ.

ABSTRACT

A trivially zero minor of a matrix is a minor having all its terms in the Leibniz formula equal to zero. A matrix is superregular if all of its minors that are not trivially zero are nonzero. In the area of Coding Theory, superregular matrices over finite fields are connected with codes with optimum distance proprieties. When a superregular matrix has all its entries nonzero, it is called full superregular and these matrices are used to construct Maximum Distance Separable block codes. In the context of convolutional codes, lower triangular Toeplitz superregular matrices are employed to build convolutional codes with optimal column distance. Although full superregular matrices over small fields are known (e.g. Cauchy matrices), the few known general constructions of these matrices having a lower triangular Toeplitz structure require very large field sizes. In this work we investigate lower triangular Toeplitz superregular matrices over small finite prime fields. Following the work of Hutchinson, Smarandache and Trumpf, we study the minimum number of different nontrivial minors that such a matrix have, and exhibit concrete constructions of superregular matrices of this kind.

Motivation & Objective

• To investigate the existence and construction of lower triangular Toeplitz superregular matrices over small finite prime fields.
• To determine the minimum number of distinct nontrivial minors such matrices can have.
• To provide explicit constructions of superregular matrices for γ = 8 and γ = 9 over small prime fields.
• To explore the feasibility of constructing such matrices for γ = 10 using computational search and random sampling.
• To propose conjectures on field size requirements and minor diversity for general γ.

Proposed method

• Constructing lower triangular Toeplitz matrices Aγ with constant diagonals defined by a vector (a₁, a₂, ..., aγ−1).
• Using a greedy algorithm in Maple to search for coefficient vectors that yield superregular matrices over small finite fields.
• Applying constraints that aᵢ ∉ Sᵢ at each step to avoid trivial minors, based on prior recursive conditions.
• Performing exhaustive or random search over small prime fields F_p for γ = 8, 9, and 10 to find valid coefficient vectors.
• Computing all minors of the matrix and verifying that all nontrivial minors are nonzero to confirm superregularity.
• Analyzing the number of distinct nontrivial minors and their distribution across different field sizes.

Experimental results

Research questions

• RQ1 What is the minimum number of distinct nontrivial minors that a lower triangular Toeplitz superregular matrix of order γ can have over a finite prime field?
• RQ2 For which small prime fields F_p does there exist a lower triangular Toeplitz superregular matrix of order γ = 10?
• RQ3 Can superregular matrices of order γ = 8 and γ = 9 be explicitly constructed over small finite prime fields?
• RQ4 What field size is sufficient to guarantee the existence of such superregular matrices for a given γ?
• RQ5 Are there general patterns or bounds on the number of distinct minors as a function of γ and p?

Key findings

• The authors successfully constructed lower triangular Toeplitz superregular matrices of order 8 and 9 over the finite prime fields F_173 and F_193.
• For γ = 10, they found superregular matrices over F_173, F_193, F_199, F_227, F_229, F_239, F_251, and F_257, with relative success frequencies increasing with field size.
• The number of distinct nontrivial minors in the γ = 10 matrix over F_257 reached 256 out of a possible 257, indicating near-maximal diversity.
• The study confirms that superregular matrices of order 10 exist over F_p for p ≥ 173, with increasing success rates as p increases.
• The authors conjecture that for any γ ≥ 2 and odd prime p, there exists a vector (a₁, ..., aγ−1) such that the number of distinct minors is at most Nγ/2 + 2.
• They further conjecture that for γ ≥ 2, a lower triangular Toeplitz superregular matrix exists over any finite field F with |F| ≥ 2^(2γ/3).