[Paper Review] The spectra of generalized Paley graphs and their associated irreducible cyclic codes.

Ricardo A. Podestá, Denis E. Videla|arXiv (Cornell University)|Aug 21, 2019

Coding theory and cryptography15 references4 citations

TL;DR

This paper establishes a direct correspondence between the spectra of generalized Paley graphs Γ(k,q) and their associated irreducible p-ary cyclic codes 𝒞(k,q), showing that the eigenvalues of the graph and the weight distribution of the code mutually determine each other under the condition k ∣ (q−1)/(p−1). It explicitly computes the spectra in the semiprimitive case using Gaussian periods and applies the results to reduce the number of rational points in Artin-Schreier curves and to compute Gaussian periods.

ABSTRACT

For $q=p^m$ with $p$ prime and $k\mid q-1$, we consider the generalized Paley graph $\Gamma(k,q) = Cay(\mathbb{F}_q, R_k)$, with $R_k=\{ x^k : x \in \mathbb{F}_q^* \}$, and the irreducible $p$-ary cyclic code $\mathcal{C}(k,q) = \{( extrm{Tr}_{q/p}(\gamma \omega^{ik})_{i=0}^{n-1})\}_{\gamma \in \mathbb{F}_q}$, with $\omega$ a primitive element of $\mathbb{F}_q$ and $n= frac{q-1}{k}$. We compute the spectra of $\Gamma(k,q)$ in terms of Gaussian periods and give $Spec(\Gamma(k,q))$ explicitly in the semiprimitive case. We then show that the spectra of $\Gamma(k,q)$ and $\mathcal{C}(k,q)$ are mutually determined by each other if further $k\mid frac{q-1}{p-1}$. Also, we use known characterizations of generalized Paley graphs which are cartesian decomposable to explicitly compute the spectra of the corresponding associated irreducible cyclic codes. As applications, we give reduction formulas for the number of rational points in Artin-Schreier curves and to the computation of Gaussian periods.

Motivation & Objective

• To establish a mutual determination between the spectra of generalized Paley graphs and their associated irreducible cyclic codes.
• To compute the eigenvalue spectrum of Γ(k,q) explicitly in the semiprimitive case using Gaussian periods.
• To apply spectral results to derive reduction formulas for the number of rational points in Artin-Schreier curves.
• To use known characterizations of cartesian-decomposable generalized Paley graphs to compute spectra of their associated codes.

Proposed method

• Define the generalized Paley graph Γ(k,q) as a Cayley graph over 𝔽_q with connection set R_k = {x^k : x ∈ 𝔽_q^*} for k ∣ q−1.
• Represent the irreducible cyclic code 𝒞(k,q) using trace functions Tr_{q/p}(γω^{ik}) for γ ∈ 𝔽_q and ω a primitive element of 𝔽_q.
• Express the spectrum of Γ(k,q) in terms of Gaussian periods, particularly in the semiprimitive case where the multiplicative order condition holds.
• Establish a duality between the eigenvalues of the graph and the weight distribution of the code under the condition k ∣ (q−1)/(p−1).
• Leverage known results on cartesian decomposability of generalized Paley graphs to derive explicit spectral formulas for their associated codes.
• Apply the spectral results to reduce the number of rational points in Artin-Schreier curves and to compute Gaussian periods.

Experimental results

Research questions

• RQ1 How are the spectra of generalized Paley graphs Γ(k,q) related to the weight distribution of their associated irreducible cyclic codes 𝒞(k,q)?
• RQ2 What is the explicit form of the spectrum of Γ(k,q) in the semiprimitive case?
• RQ3 Under what conditions does the spectrum of the graph and the code mutually determine each other?
• RQ4 How can spectral properties of generalized Paley graphs be used to reduce the number of rational points in Artin-Schreier curves?
• RQ5 Can known structural properties like cartesian decomposability of generalized Paley graphs lead to explicit computation of associated code spectra?

Key findings

• The spectrum of the generalized Paley graph Γ(k,q) is fully determined by Gaussian periods, with an explicit formula provided in the semiprimitive case.
• When k divides (q−1)/(p−1), the spectra of Γ(k,q) and the associated code 𝒞(k,q) are mutually determined: knowledge of one fully determines the other.
• The spectral results enable reduction formulas for the number of rational points in Artin-Schreier curves over finite fields.
• For cartesian-decomposable generalized Paley graphs, the spectra of the associated irreducible cyclic codes are explicitly computed via the graph's spectral structure.
• The framework provides a new method for computing Gaussian periods using spectral data from generalized Paley graphs.
• The results establish a deep algebraic link between graph theory, coding theory, and arithmetic geometry over finite fields.