Non-linear Maximum Rank Distance Codes Constructed from Cones over Exterior Sets in Finite Projective Spaces

Core Concepts

A new class of non-linear maximum rank distance (MRD) codes is constructed using a geometric approach involving cones over exterior sets in finite projective spaces.

Abstract

The article presents a geometric construction of a new family of non-linear MRD codes. The key insights are: The codes are obtained from a cone in the finite projective space PG(n-1, q^n), where the vertex of the cone is a proper subspace and the base is a special point set that is skew with the vertex. The base of the cone is linked to non-linear MRD codes constructed in prior work, which can be obtained by puncturing the codes in the new family. It is shown that this new class of codes is not equivalent to the non-linear MRD codes constructed by Otal and Özbudak. The construction involves the following steps: Defining an (n-k+1)-embedding Γ of the canonical subgeometry Σ in PG(n-1, q^n). Identifying a maximum exterior set E with respect to Ω_n-k-1(Γ). Constructing the cone K(Λ^, E), where Λ^ is a (k-2)-dimensional subspace disjoint from Σ. Proving that K(Λ^*, E) is a maximum exterior set with respect to Ω_n-k-1(Σ). Showing that the resulting non-linear MRD code Cσ,T, where T is a subset of F_q^*, contains the codes constructed in prior work as punctured versions.

Stats

The size of the exterior set E is given as: |E| ≤ (q^(n-h-1) - 1) / (q - 1), where 0 ≤ h ≤ n-1. The size of the cone K(Λ^, E) is given as: |K(Λ^, E)| = (q^nk - 1) / (q - 1).

Quotes

"The set X is the Cσ_F-set with vertices A and B generated by a σ-collineation Φ between the star of lines through A and B contained in Λ." "The punctured code C^[k-2]_σ,T ⊂ F^((n-k+2)×n)_q obtained from Cσ,T by deleting the last (k-2) rows is exactly the code constructed in [8, Theorem 5.1], while for k = 2 the code appeared in [9] and for n = 3, k = 2 in [2]."

Key Insights Distilled From

Non-linear MRD codes from cones over exterior sets

by Nicola Duran... at arxiv.org 05-03-2024

https://arxiv.org/pdf/2305.19027.pdf

Deeper Inquiries

How can the geometric properties of the cone construction be further exploited to analyze the structure and properties of the resulting non-linear MRD codes

The geometric properties of the cone construction can be further exploited to analyze the structure and properties of the resulting non-linear MRD codes in several ways. Firstly, the concept of cones over exterior sets can provide insights into the rank distribution and minimum distance properties of the codes. By studying the intersection of the cone with different subspaces and the behavior of lines passing through points in the cone, we can gain a deeper understanding of the code's structure. Additionally, the geometric construction allows for the identification of maximum exterior sets with respect to specific secant varieties, which can lead to the creation of non-linear MRD codes with optimal size and distance properties. Analyzing the properties of these exterior sets and their relationship to the underlying projective space can provide valuable information about the code's performance and error-correcting capabilities. Furthermore, the geometric approach can help in characterizing the inequivalence of the constructed codes with respect to other known non-linear MRD codes. By examining the geometric configurations of the cones and their bases, we can determine the uniqueness and distinctiveness of the codes in terms of their structural properties.

Can the techniques used in this work be extended to construct non-linear MRD codes for other parameter settings beyond the ones considered here

The techniques used in this work can be extended to construct non-linear MRD codes for a wide range of parameter settings beyond the ones considered in the current study. By exploring different combinations of subspaces, embeddings, and exterior sets in projective spaces of varying dimensions and field sizes, new classes of non-linear MRD codes can be generated. For instance, the concept of embeddings and cones over exterior sets can be applied to construct non-linear MRD codes for different values of n, q, and d. By adapting the geometric construction to different geometries and subspaces, it is possible to create a diverse set of codes with unique properties and performance characteristics. Moreover, the approach of utilizing exterior sets with respect to specific secant varieties can be extended to explore non-linear MRD codes in other algebraic and geometric structures. By investigating the geometric properties of different sets and their relationships within projective spaces, novel constructions of non-linear MRD codes can be developed for various applications and scenarios.

What are the potential applications of these non-linear MRD codes in areas such as network coding, cryptography, or space-time coding

The non-linear MRD codes constructed using the techniques described in this work have several potential applications in areas such as network coding, cryptography, and space-time coding. In network coding, these codes can be utilized for efficient error correction and data recovery in communication networks. The unique properties of non-linear MRD codes, such as their high minimum distance and error-correcting capabilities, make them suitable for ensuring reliable and secure data transmission in networked systems. In cryptography, non-linear MRD codes can be employed for secure data encryption and decryption processes. The strong error-correcting abilities of these codes can enhance the security and reliability of cryptographic systems, protecting sensitive information from unauthorized access and tampering. In space-time coding, non-linear MRD codes can be used for improving the performance of wireless communication systems. By encoding data using these codes, it is possible to achieve higher data rates, improved spectral efficiency, and enhanced reliability in space-time communication scenarios, such as multiple-input multiple-output (MIMO) systems. Overall, the applications of non-linear MRD codes are diverse and versatile, offering valuable solutions for various challenges in modern communication, security, and coding systems.