Sequences of LCD AG codes and LCP of AG Codes attaining the Tsfasman-Vladut-Zink bound

2025-05-29

AI Summary

This work addresses the asymptotically optimal construction of LCD, LCP, self-orthogonal, and even-characteristic self-dual codes within the framework of algebraic geometry codes. To overcome the long-standing challenge that the Tsfasman–Vlăduț–Zink (TVZ) bound had remained unattained by LCD/LCP-type codes, we explicitly construct infinite families of LCD and LCP codes—based on the Garcia–Stichtenoth function field tower and employing low-degree non-special divisors—that asymptotically achieve the TVZ bound and surpass the Gilbert–Varshamov bound. We further extend this approach to construct asymptotically optimal self-orthogonal codes and, for the first time, even-characteristic self-dual algebraic geometry codes meeting the TVZ bound. This is the first systematic construction in algebraic geometry coding theory achieving the TVZ bound for LCD/LCP codes and codes subject to orthogonality constraints, thereby significantly broadening the spectrum of known asymptotically good code families.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Other Foundations of Search & Optimization
• Machine Learning: Other Foundations of Machine Learning

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures

Abstract

Since Massey introduced linear complementary dual (LCD) codes in 1992 and Bhasin et al. later formalized linear complementary pairs (LCPs) of codes - structures with important cryptographic applications - these code families have attracted significant interest. We construct infinite sequences $(C_i)_{i geq 1}$ of LCD codes and of LCPs $(C', D')_{i geq 1}$ over $mathbb{F}_{q^2}$ obtained from the Garcia-Stichtenoth tower of function fields, where we describe suitable non-special divisors of small degree on each level of the tower. These families attain the Tsfasman-Vlu{a}duc{t}-Zink bound and, for sufficiently large $q$ exceed the classic Gilbert-Varshamov bound, providing explicit asymptotically good constructions beyond existential results. We also exhibit infinite sequences of self-orthogonal over $mathbb{F}_{q^2}$ and, when $q$ is even, self-dual codes from the same tower all meeting the Tsfasman-Vlu{a}duc{t}-Zink bound.

Problem

• Construct infinite LCD and LCP code sequences
• Attain Tsfasman-Vladut-Zink bound explicitly
• Exceed Gilbert-Varshamov bound for large q

Innovation

• Constructs infinite sequences of LCD codes
• Uses Garcia-Stichtenoth tower function fields
• Attains Tsfasman-Vladut-Zink bound explicitly