[Paper Review] On the Constructions of MDS Self-dual Codes via Cyclotomy

Aixian Zhang, Keqin Feng|arXiv (Cornell University)|Nov 13, 2019

Coding theory and cryptography8 references4 citations

TL;DR

This paper presents a novel construction of MDS self-dual codes over finite fields using generalized Reed-Solomon (GRS) and extended GRS codes, where the evaluation sets are unions of cyclotomic classes of the multiplicative group of the field. By expressing self-duality conditions in terms of cyclotomic numbers, the authors derive new series of MDS self-dual codes with previously unknown lengths, particularly for odd prime powers and various orders of cyclotomy.

ABSTRACT

MDS self-dual codes over finite fields have attracted a lot of attention in recent years by their theoretical interests in coding theory and applications in cryptography and combinatorics. In this paper we present a series of MDS self-dual codes with new length by using generalized Reed-Solomon codes and extended generalized Reed-Solomon codes as the candidates of MDS codes and taking their evaluation sets as an union of cyclotomic classes. The conditions on such MDS codes being self-dual are expressed in terms of cyclotomic numbers.

Motivation & Objective

• To construct new MDS self-dual codes over finite fields with previously unknown lengths.
• To generalize prior constructions that used subgroups or subspaces as evaluation sets by employing unions of cyclotomic classes.
• To establish necessary and sufficient conditions for MDS codes from GRS and EGRS codes to be self-dual using cyclotomic number theory.
• To extend the applicability of cyclotomic number theory to MDS self-dual code construction beyond the semiprimitive case.
• To provide explicit families of MDS self-dual codes for various orders of cyclotomy, especially for e = 4 and e = 2.

Proposed method

• The construction uses generalized Reed-Solomon (GRS) and extended GRS (EGRS) codes as MDS code candidates.
• The evaluation set S is chosen as a union of cosets of a subgroup D = ⟨θ^e⟩ in F_q^*, where F_q^* = ⟨θ⟩.
• Self-duality is determined by conditions on cyclotomic numbers, particularly the parity of (odd, I−i) sums over selected index sets I.
• The method leverages known results on cyclotomic numbers of order e = 2 and e = 4, especially their parity and algebraic structure.
• The authors apply Theorem 3.3 to translate conditions on cyclotomic number parity into existence criteria for self-dual codes.
• Specific constructions are derived for e = 2 and e = 4, with detailed analysis based on the prime power q modulo 4, 8, and 16.

Experimental results

Research questions

• RQ1 For which finite fields F_q and unions of cyclotomic classes can MDS self-dual codes be constructed via GRS and EGRS codes?
• RQ2 What are the necessary and sufficient conditions on cyclotomic numbers for such MDS codes to be self-dual?
• RQ3 How do the parameters of the resulting MDS self-dual codes (length, field size) depend on the order e of the cyclotomy and the structure of the index set I?
• RQ4 Can the construction yield new code lengths not previously known in the literature?
• RQ5 What role does the quadratic character and the decomposition q = s² + 4t² play in determining the existence of such codes?

Key findings

• For e = 2, if q ≡ 1 (mod 4), then 2f + 2 ∈ Σ(eg, q); if q ≡ 1 (mod 8), then f + 2 ∈ Σ(eg, q).
• If q ≡ 3 (mod 4), then 2f + 2 ∈ Σ(eg, q); if q ≡ 7 (mod 8), then f + 1 ∈ Σ(eg, q) ∩ Σ(g, q).
• For e = 4 and p ≡ 1 (mod 4), if 2 | f (i.e., q ≡ 1 (mod 8)), then f, 2f ∈ Σ(g, q) and f + 2, 2f + 2, 4f + 2 ∈ Σ(eg, q).
• When q ≡ 1 (mod 16) and 4 | t, or q ≡ 9 (mod 16) and t ≡ 2 (mod 4), then 3f ∈ Σ(g, q) and 3f + 2 ∈ Σ(eg, q).
• For p ≡ 3 (mod 4), if p ≡ 7 (mod 8) or p ≡ 3 (mod 8) with 2 | m, then f, 2f ∈ Σ(g, q) and fl + 2 ∈ Σ(eg, q) for 1 ≤ l ≤ 4.
• In the semiprimitive case with p ≡ 3 (mod 8) and 2 ∤ m, fl ∈ Σ(g, q) for 1 ≤ l ≤ 3 and fl + 2 ∈ Σ(eg, q) for l = 1, 2, 4.