[Paper Review] Transitive and Self-dual Codes Attaining the Tsfasman-Vladut-Zink Bound

Henning Stichtenoth|ArXiv.org|Jun 14, 2005

Coding theory and cryptography4 citations

TL;DR

This paper introduces transitive codes—linear codes with transitive automorphism groups—as a generalization of cyclic codes and proves they achieve the Tsfasman-Vladut-Zink (TVZ) bound over finite fields 𝔽_q when q = ℓ² is a square. Using a new asymptotically optimal Galois tower of function fields, the authors construct sequences of transitive, self-orthogonal, and self-dual codes that asymptotically meet the TVZ bound, significantly improving upon the Gilbert-Varshamov bound for q ≥ 49.

ABSTRACT

We introduce - as a generalization of cyclic codes - the notion of transitive codes, and we show that the class of transitive codes is asymptotically good. Even more, transitive codes attain the Tsfasman-Vladut-Zink bound over F_q, for all aquares q=l^2. We also show that self-orthogonal and self-dual codes attain the Tsfasman-Vladut-Zink bound, thus improving previous results about self-dual codes attaining the Gilbert-Varshamov bound. The main tool is a new asymptotically optimal tower (E_n) of function fields over F_q where all extensions E_n/E_0 are Galois.

Motivation & Objective

• To resolve the open problem of whether cyclic codes are asymptotically good by introducing a broader class of transitive codes.
• To prove that transitive codes over 𝔽_q with q = ℓ² attain the Tsfasman-Vladut-Zink (TVZ) bound, thus establishing their asymptotic goodness.
• To show that self-orthogonal and self-dual codes also achieve the TVZ bound, improving prior results that only reached the Gilbert-Varshamov bound.
• To construct a new asymptotically optimal tower of function fields over 𝔽_q (q=ℓ²) with all extensions Galois, enabling the construction of codes with optimal parameters.

Proposed method

• Introduce transitive codes as a generalization of cyclic codes, defined by transitive automorphism groups in the symmetric group S_n.
• Construct a new tower of function fields E₀ ⊆ E₁ ⊆ E₂ ⊆ … over 𝔽_q (q=ℓ²) where all extensions E_n/E₀ are Galois and have maximal class number growth.
• Use geometric Goppa codes derived from divisors and differentials on the function fields in the tower to generate linear codes with controlled parameters.
• Apply the residue of a differential form at rational places to characterize the dual of the constructed codes, enabling self-orthogonality and self-duality.
• Leverage the Galois action on places to prove transitivity of the codes, ensuring the automorphism group acts transitively on the coordinate positions.
• Use the Drinfeld-Vladut bound and the genus-growth properties of the tower to derive the asymptotic parameters and verify that the codes meet the TVZ bound.

Experimental results

Research questions

• RQ1 Can the class of transitive codes—generalizing cyclic codes—achieve the Tsfasman-Vladut-Zink bound over finite fields of square order?
• RQ2 Do self-orthogonal and self-dual codes over 𝔽_q (q=ℓ²) attain the TVZ bound, thereby improving upon the Gilbert-Varshamov bound?
• RQ3 Is there a new family of function fields over 𝔽_q (q=ℓ²) with all Galois extensions that enables the construction of asymptotically optimal codes?
• RQ4 Can geometric Goppa codes from such a tower be shown to be transitive, self-orthogonal, or self-dual via differential residue analysis?
• RQ5 What is the asymptotic trade-off between rate R and relative minimum distance δ for these structured codes, and how does it compare to known bounds?

Key findings

• Transitive codes over 𝔽_q with q=ℓ² achieve the Tsfasman-Vladut-Zink bound, with asymptotic rate R ≥ 1−δ−1/(ℓ−1) for δ∈(0,1−1/(ℓ−1)).
• Self-orthogonal codes over 𝔽_q (q=ℓ²) attain the TVZ bound, with asymptotic rate R ≥ 1−δ−1/(ℓ−1) for δ∈(0,1−1/(ℓ−1)) and R≤1/2.
• Self-dual codes over 𝔽_q (q=ℓ²) also achieve the TVZ bound, with asymptotic rate R=1/2 and relative minimum distance δ satisfying R=1−δ−1/(ℓ−1).
• The constructed geometric Goppa codes C^{(n)}_{a,b} are transitive and iso-orthogonal for 0≤a≤a_n/2 and 0≤b≤b_n/2, and iso-dual for a=a_n/2, b=b_n/2.
• The dual of C^{(n)}_{a,b} is given by (C^{(n)}_{a,b})^⊥ = u·C^{(n)}_{a_n−a,b_n−b}, where u is the vector of residues of a differential form at rational places.
• For q≥49, the TVZ bound strictly improves upon the Gilbert-Varshamov bound, and the constructed codes meet this improved bound, demonstrating their superiority.