The Cayley graphs associated with some quasi-perfect Lee codes are Ramanujan graphs

The paper titled "The Cayley graphs associated with some quasi-perfect Lee codes are Ramanujan graphs" by Khodakhast Bibak, Bruce M. Kapron, and Venkatesh Srinivasan provides a solution to a conjecture posed by Camarero and Martínez in their 2016 paper published in IEEE Transactions on Information Theory. The conjecture pertains to the Ramanujan property of the Cayley graph $\mathcal{G}_p$ induced by the set of units $S_2$ of the ring of Gaussian integers modulo a prime number $p\equiv \pm 5 \pmod{12}$, denoted as $\Z_p[i]$. Camarero and Martínez had shown that $\mathcal{G}_p$ induces a 2-quasi-perfect Lee code over $\Z_p^m$, where $m=2\lfloor \frac{p}{4}\rfloor$. In this paper, the authors prove that $\mathcal{G}_p$ is indeed a Ramanujan graph for every prime $p$ such that $p\equiv 3 \pmod{4}$. This result is obtained using Deligne's bound from 1977 for estimating a particular kind of trigonometric sum and Lovász's result from 1975 (or Babai's result from 1979) which gives the eigenvalues of Cayley graphs of finite Abelian groups. The proof techniques used in this paper may motivate further work in the interactions between spectral graph theory, character theory, and coding theory. Additionally, these techniques may provide new ideas towards resolving the famous Golomb-Welch conjecture on the existence of perfect Lee codes. Overall, this paper contributes to our understanding of Ramanujan graphs and their properties. It highlights how existing mathematical tools can be leveraged to solve new problems and underscores the potential for interdisciplinary collaborations between different areas of mathematics such as spectral graph theory, character theory, and coding theory.

The Cayley Graphs Associated with Some Quasi-Perfect Lee Codes are Ramanujan Graphs

In 2016, Camarero and Martínez published a paper in IEEE Transactions on Information Theory which posed the conjecture that the Cayley graph $\mathcal{G}_p$ induced by the set of units $S_2$ of the ring of Gaussian integers modulo a prime number $p\equiv \pm 5 \pmod{12}$, denoted as $\Z_p[i]$, has the Ramanujan property. This conjecture was recently proven to be true by Khodakhast Bibak, Bruce M. Kapron, and Venkatesh Srinivasan in their 2020 paper titled “The Cayley graphs associated with some quasi-perfect Lee codes are Ramanujan graphs”. In this article, we will discuss what this research paper contributes to our understanding of Ramanujan graphs and how it highlights potential for interdisciplinary collaborations between different areas of mathematics such as spectral graph theory, character theory, and coding theory.

Background

A graph is said to have the Ramanujan property if its second largest eigenvalue (in absolute value) is bounded above by 2√(d−1), where d is its degree. The study of these types of graphs began in 1993 when Lubotzky et al showed that certain expander families had this property. Since then, many researchers have studied these so-called “Ramanujan graphs” due to their numerous applications in theoretical computer science such as network design and cryptography.

In addition to being an interesting object from a purely mathematical standpoint, there has been recent interest in studying Ramanujan graphs from a coding theoretic perspective due to their connection with perfect Lee codes over finite fields. A perfect Lee code over a finite field $\mathbb{F}_{q^m}$ is defined as an injective map from $\mathbb{F}_{q^m}\rightarrow \mathbb{F}_{q^m}\times\mathbb{F}_{q^m}\times\cdots\times\mathbb{F}_{q^m}, m$ times which preserves Hamming distances between codewords (i.e., it maps two distinct elements x and y into two distinct vectors whose Hamming distance equals d(x,y)). Perfect Lee codes exist only when q=4 or q=9; however they do not exist for all values of m even when q=4 or 9 (see Golomb-Welch Conjecture).

Camarero & Martínez's Conjecture

Camarero & Martínez conjectured that for every prime number p such that $p\equiv \pm 5 \pmod {12},$ the Cayley graph $\mathcal G_p$ induced by the set of units $S_2$of $\Z_p[i]$, where $S_2 = \{\pm 1 , i , -i , p+1 , -(p+1)\},$ has the Ramanujan property. They also showed that if this were true then it would imply that for every such prime p there exists a 2-quasi-perfect Lee code over $\Z_p^{2\lfloor \frac {p}{4}\rfloor}.$

Khodakhast Bibak et al.'s Result

Khodakhast Bibak et al., proved Camarero & Martínez's conjecture using Deligne's bound from 1977 for estimating a particular kind of trigonometric sum and Lovász's result from 1975 (or Babai's result from 1979) which gives us information about eigenvalues of Cayley graphs associated with Abelian groups . Specifically they show that if p satisfies $p≡ 3mod 4$, then indeed $\mathcal G_P$ has the desired properties making it a valid example for further exploration into connections between spectral graph theory character theory and coding theory .

Implications

This research provides new insight into existing problems related to perfect Lee codes over finite fields . It also demonstrates how existing mathematical tools can be leveraged towards solving new problems related to interactions between spectral graph theory character theory and coding theory . Additionally this work may provide new ideas towards resolving Golomb Welch Conjecture on existence/non existence perfect lee codes . Overall this paper contributes significantly towards our understanding about ramunajan graphes properties thus highlighting potential interdisciplinary collaborations between different areas mathematics like spectral graph theoriy character theoriy coding theoriy etc..