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

TL;DR

This paper constructs infinite sequences of LCD codes and LCPs over finite fields from algebraic geometry, achieving the Tsfasman-Vlu-Zink bound and surpassing classical bounds, with applications in cryptography.

Contribution

It introduces explicit asymptotically good sequences of LCD codes and LCPs from Garcia-Stichtenoth towers, attaining the Tsfasman-Vlu-Zink bound and exceeding the Gilbert-Varshamov bound.

Findings

Sequences attain the Tsfasman-Vlu-Zink bound.

Sequences exceed the Gilbert-Varshamov bound for large q.

Existence of infinite sequences of self-orthogonal and self-dual codes meeting the bound.

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-Vl\u{a}du\c{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-Vl\u{a}du\c{t}-Zink bound.