arXiv subjects

Shohei Satake

Publications and source records attributed to Shohei Satake.

13 recordsLinked to original sources

arXiv2026

$2$-quasi-perfect Lee codes and abelian Ramanujan graphs: a new construction and relationship

This paper presents a new explicit infinite family of 2-quasi-perfect $p$-ary Lee codes of length $\frac{q-1}{2}$ and dimension $\frac{q-1}{2}-2k$ for $q = p^k \ge 14$, $p\geq 5$ a prime. Our codes are derived from the generating set $H_q = \{(a, a^3) \mid a \in \mathbb{F}_q^*\}$ of the additive group of the finite field $\mathbb{F}_{q^2}$. Furthermore, we bridge between 2-quasi-perfect Lee codes constructed by Mesnager, Tang, and Qi and well-known abelian Ramanujan graphs, specifically Li's graphs and finite Euclidean graphs, providing a unified theoretical framework for these families.

cs.IT↗

arXiv2025

Topological properties of generalized Markoff mod $p$ graphs

The generalized Markoff mod $p$ graph is defined via the equation $x^2+y^2+z^2=xyz+\kappa$ over the finite field $\mathbb{F}_p$ of prime order $p$. In this paper, we investigate the topological properties of the graph such as non-planarity, surface embeddability, and the existence of short cycles. Our approach is based on a systematic construction of $K_{3,3}$-subdivisions, integrating techniques from graph theory, computer algebra, and number theory.

math.NT↗

arXiv2025

On cyclotomic nearly-doubly-regular tournaments

Nearly-doubly-regular tournaments have played significant roles in extremal graph theory. In this note, we construct new cyclotomic nearly-doubly-regular tournaments and determine their spectrum by establishing a new connection between cyclotomic nearly-doubly-regular tournaments and almost difference sets from combinatorial design theory. Furthermore, under the celebrated Hardy-Littlewood conjecture F in analytic number theory, our results confirm the conjecture due to Sergey Savchenko (J. Graph Theory {\bf 83} (2016), 44--77) on the existence of infinitely many nearly-doubly-regular tournaments with the canonical spectrum.

math.CO↗

arXiv2024

On the Paley RIP and Paley graph extractor

Constructing explicit RIP matrices is an open problem in compressed sensing theory. In particular, it is quite challenging to construct explicit RIP matrices that break the square-root bottleneck. On the other hand, providing explicit $2$-source extractors is a fundamental problem in theoretical computer science, cryptography and combinatorics. Nowadays, there are only a few known constructions for explicit $2$-source extractors (with negligible errors) that break the half barrier for min-entropy. In this paper, we establish a new connection between RIP matrices breaking the square-root bottleneck and $2$-source extractors breaking the half barrier for min-entropy. Here we focus on an RIP matrix (called the Paley ETF) and a $2$-source extractor (called the Paley graph extractor), where both are defined from quadratic residues over the finite field of odd prime order $p\equiv 1 \pmod{4}$. As a main result, we prove that if the Paley ETF breaks the square-root bottleneck, then the Paley graph extractor breaks the half barrier for min-entropy as well. Since it is widely believed that the Paley ETF breaks the square-root bottleneck, our result accordingly provides a new affirmative intuition on the conjecture for the Paley graph extractor by Benny Chor and Oded Goldreich.

math.CO↗

arXiv2022

Private Simultaneous Messages Based on Quadratic Residues

Private Simultaneous Messages (PSM) model is a minimal model for secure multiparty computation. Feige, Kilian, and Naor (STOC 1994) and Ishai (Cryptology and Information Security Series 2013) constructed PSM protocols based on quadratic residues. In this paper, we define QR-PSM protocols as a generalization of these protocols. A QR-PSM protocol is a PSM protocol whose decoding function outputs the quadratic residuosity of what is computed from messages. We design a QR-PSM protocol for any symmetric function $f: \{0,1\}^n \rightarrow \{0,1\}$ of communication complexity $O(n^2)$. As far as we know, it is the most efficient PSM protocol since the previously known best PSM protocol was of $O(n^2\log n)$ (Beimel et al., CRYPTO 2014). We also study the sizes of the underlying finite fields $\mathbb{F}_p$ in the protocols since the communication complexity of a QR-PSM protocol is proportional to the bit length of the prime $p$. In particular, we show that the $N$-th Peralta prime $P_N$, which is used for general QR-PSM protocols, can be taken as at most $(1+o(1))N^2 2^{2N-2}$, which improves the Peralta's known result (Mathematics of Computation 1992) by a constant factor $(1+\sqrt{2})^2$.

cs.CR↗

arXiv2021

On high-girth expander graphs with localized eigenvectors

The main purpose of this paper is to construct high-girth regular expander graphs with localized eigenvectors for general degrees, which is inspired by a recent work due to Alon, Ganguly and Srivastava (to appear in Israel J. Math.).

math.CO↗

arXiv2020

On the restricted isometry property of the Paley matrix

In this paper, we prove that the Paley graph conjecture implies that the Paley matrix has restricted isometry property (RIP) beating the square-root bottleneck for the sparsity level. Moreover, we show that the RIP of the Paley matrix implies an improved bound on the size of transitive subtournaments in the Paley tournament.

math.CO↗

arXiv2020

On Compressed Sensing Matrices Breaking the Square-Root Bottleneck

Compressed sensing is a celebrated framework in signal processing and has many practical applications. One of challenging problems in compressed sensing is to construct deterministic matrices having restricted isometry property (RIP). So far, there are only a few publications providing deterministic RIP matrices beating the square-root bottleneck on the sparsity level. In this paper, we investigate RIP of certain matrices defined by higher power residues modulo primes. Moreover, we prove that the widely-believed generalized Paley graph conjecture implies that these matrices have RIP breaking the square-root bottleneck.

cs.IT↗

arXiv2020

On $2$-parent-identifying set systems of block size $4$

Parent-identifying set system is a kind of combinatorial structures with applications to broadcast encryption. In this paper we investigate the maximum number of blocks $I_2(n,4)$ in a $2$-parent-identifying set system with ground set size $n$ and block size $4$. The previous best known lower bound states that $I_2(n,4)=Ω(n^{4/3+o(1)})$. We improve this lower bound by showing that $I_2(n,4)= Ω(n^{3/2-o(1)})$ using techniques in additive number theory.

math.CO↗

arXiv2020

On explicit random-like tournaments

We give a new theorem describing a relation between the quasi-random property of regular tournaments and their spectra. This provides many solutions to a constructing problem mentioned by Erdős and Moon (1965) and Spencer (1985).

math.CO↗

arXiv2019

A note on the relation between two properties of random graphs

The $t$-e.c. and pseudo-random property are typical properties of random graphs. In this note, we study the gap between them which has not been studied well. As a main result, we give the first explicit construction of infinite families of t-e.c. graphs which are not families of best possible pseudo-random graphs.

math.CO↗

arXiv2019

On expander Cayley graphs from Galois rings

In this paper, we study new Cayley graphs over the additive group of Galois rings. First we prove that they are expander graphs by using a Weil-Carlitz-Uchiyama type estimation of character sums for Galois rings. We also show that Cayley graphs from Galois rings of characteristic 4 form a new infinite family of Ramanujan graphs by an elementary eigenvalue estimation. Moreover some other spectral properties of our graphs are also discussed.

math.CO↗

arXiv2019

A constructive solution to a problem of ranking tournaments

A tournament is an oriented complete graph. The problem of ranking tournaments was firstly investigated by P. Erdős and J. W. Moon. By probabilistic methods, the existence of "unrankable" tournaments was proved. On the other hand, they also mentioned the problem of explicit constructions. However, there seems to be only a few of explicit constructions of such tournaments. In this note, we give a construction of many such tournaments by using skew Hadamard difference sets which have been investigated in combinatorial design theory.

math.CO↗