Constructing Pseudo-Random and de Bruijn Array Codes from Irreducible Polynomials

Core Concepts

This paper introduces the concepts of pseudo-random array codes and de Bruijn array codes, which are two-dimensional generalizations of one-dimensional sequences with the window property. It presents constructions for these codes based on folding sequences generated by irreducible polynomials.

Abstract

The paper starts by providing the basic definitions and results for two-dimensional arrays and array codes, including perfect maps, shortened perfect maps, pseudo-random arrays, and their corresponding codes. It then presents a construction for pseudo-random array codes based on folding a sequence generated by an irreducible polynomial that is not primitive. The key idea is to use Theorem 6, which provides a necessary and sufficient condition for the folding to yield a pseudo-random array. Next, the paper discusses two-dimensional de Bruijn array codes, which are the two-dimensional analogs of perfect factors in the de Bruijn graph. Several direct and recursive constructions for these codes are presented, including Theorems 8-16. The paper concludes by highlighting open problems and directions for future research, such as: Providing new constructions for shortened de Bruijn arrays and pseudo-random arrays with larger window sizes Proving the sufficiency of the necessary conditions for the existence of de Bruijn array codes Exploring more constructions for zero factors and de Bruijn array codes Investigating when folding can successfully yield pseudo-random array codes beyond the parameters given in Theorem 7.

Stats

Each nonzero n-tuple appears as a row in the matrix T if and only if the columns of T are linearly independent. The set polynomial gR(x) is not divisible by the irreducible polynomial f(x) if and only if the set of R coordinates in the sequence generated by f(x) contains each nonzero n-tuple. The minimum distance of the pseudo-random array code is the weight of the sequence of the smallest weight generated by the irreducible polynomial f(x).

Quotes

"Pseudo-random arrays and perfect maps are the two-dimensional analogs of M-sequences and de Bruijn sequences, respectively." "A factor in a graph is a set of vertex disjoint cycles that contain all the vertices in the graph." "A perfect factor PF(n, k) in Gn is a set of 2n−k vertex-disjoint cycles of length 2k in Gn." "A zero factor ZF(n, k) with exponent k in Gn is a set of d vertex-disjoint cycles of length k in Gn, which contains each nonzero n-tuple exactly once as a window in one of the cycles."

Key Insights Distilled From

Pseduo-Random and de Bruijn Array Codes

by Tuvi Etzion at arxiv.org 04-12-2024

https://arxiv.org/pdf/2311.04451.pdf

Deeper Inquiries

How can the constructions for pseudo-random array codes and de Bruijn array codes be extended to larger window sizes, i.e., n × m matrices where n ≥ 3?

In order to extend the constructions for pseudo-random array codes and de Bruijn array codes to larger window sizes, specifically n × m matrices where n is greater than or equal to 3, several approaches can be considered. One way to achieve this extension is by adapting the folding technique used in the constructions to accommodate the larger window sizes. This may involve modifying the folding process to handle the increased dimensions of the arrays while ensuring that each n × m matrix appears exactly once as a window in the constructed arrays. Additionally, the choice of irreducible polynomials and the properties of the sequences generated from these polynomials play a crucial role in the construction of these array codes. By carefully selecting appropriate irreducible polynomials of higher degrees, it may be possible to generate sequences that can be folded into larger n × m arrays while maintaining the desired properties such as pseudo-randomness or perfect mapping. Furthermore, exploring the relationships between the parameters of the irreducible polynomials, the dimensions of the arrays, and the properties of the resulting codes can provide insights into how the constructions can be adapted and extended to accommodate larger window sizes. By analyzing the impact of varying parameters on the construction process, it may be possible to derive generalized methods for constructing pseudo-random and de Bruijn array codes with larger n × m matrices.

What are the implications of the shift-and-add property of pseudo-random array codes on their error-correcting capabilities and potential applications?

The shift-and-add property of pseudo-random array codes has significant implications for their error-correcting capabilities and potential applications. This property allows for efficient error detection and correction mechanisms, making these codes suitable for use in error-prone environments such as communication systems, storage systems, and data transmission. One key implication of the shift-and-add property is that it enables the detection of errors by comparing shifted versions of the arrays. By applying shifts to the arrays and performing addition operations, discrepancies or errors in the data can be easily identified. This property enhances the error-correcting capabilities of pseudo-random array codes, making them robust against noise and interference. Moreover, the shift-and-add property facilitates error correction by enabling the reconstruction of the original data from erroneous or corrupted versions. By leveraging the shift operations and additive properties of the codes, errors can be localized and corrected, ensuring data integrity and reliability. In terms of potential applications, the shift-and-add property of pseudo-random array codes makes them suitable for various fields such as telecommunications, cryptography, and data storage. These codes can be utilized in channel coding schemes, encryption algorithms, and fault-tolerant storage systems to enhance data security, reliability, and efficiency.

Can the concepts of pseudo-random and de Bruijn array codes be further generalized to higher-dimensional arrays, and what new insights or challenges would such generalizations bring?

The concepts of pseudo-random and de Bruijn array codes can indeed be further generalized to higher-dimensional arrays, extending beyond two-dimensional structures to three-dimensional or multidimensional arrays. By transitioning to higher dimensions, new insights and challenges emerge that can impact the design, implementation, and analysis of these codes. One potential approach to generalizing pseudo-random and de Bruijn array codes to higher dimensions is to consider the extension of the folding technique to accommodate additional dimensions. This may involve folding sequences generated from higher-degree irreducible polynomials into multidimensional arrays while preserving the desired properties of pseudo-randomness or perfect mapping. The transition to higher-dimensional arrays introduces complexities related to data representation, storage, and processing. New insights may arise from exploring the interactions between the dimensions of the arrays, the properties of the sequences, and the construction methods employed. Understanding how these factors interplay in higher-dimensional settings can lead to novel applications and advancements in coding theory. Challenges associated with generalizing to higher dimensions include increased computational complexity, higher memory requirements, and the need for sophisticated algorithms to handle multidimensional data structures. Analyzing the trade-offs between efficiency, error-correcting capabilities, and code performance in higher-dimensional arrays poses a significant challenge that requires innovative solutions. Overall, the generalization of pseudo-random and de Bruijn array codes to higher dimensions opens up avenues for exploring new coding techniques, addressing complex data processing tasks, and advancing the field of multidimensional coding theory.