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Rongquan Feng

Publications and source records attributed to Rongquan Feng.

At least 19 recordsLinked to original sources
arXiv2026

New Results on Limited Magnitude Error Correcting Codes

This paper investigates the existence, construction and classification of limited magnitude error-correcting codes, with a focus on splitter sets and their connections to group splittings. We establish new nonexistence results for quasi-perfect splitter sets and provide a complete classification of quasi-perfect $B[0,3](n)$ splitter sets in both singular and nonsingular cases. Furthermore, we derive improved lower bounds for the size of maximal $B[0,3](q)$ sets by investigating Cayley graphs, where $q$ is a prime. We also provide existence criteria for perfect $B[0,6](q)$ splitter sets and quasi-perfect $B[-4,4](2p)$ sets for prime $p$. For perfect burst-correcting codes, we develop a general construction framework, and prove the existence of infinite families of $(k_2,k_1)$-limited-magnitude cyclic $b$-burst-correcting codes for $k_1+k_2\le 4$ and arbitrary burst length $b$. We further provide sufficient existence conditions for general parameters $k_1$ and $k_2$. Our results combine algebraic, combinatorial, and number-theoretic methods to advance the understanding of codes tailored for flash memory and related storage systems.

cs.IT↗
arXiv2025

On the Existence and Nonexistence of Splitter Sets

In this paper, the existence of perfect and quasi-perfect splitter sets in finite abelian groups is studied, motivated by their application in coding theory for flash memory storage. For perfect splitter sets we view them as splittings of $\mathbb{Z}_n$, and using cyclotomic polynomials we derive a general condition for the existence of such splittings under certain circumstances. We further establish a relation between $B[-k, k](q)$ and $B[-(k-1), k+1](q)$ splitter sets, and give a necessary and sufficient condition for the existence of perfect $B[-1, 5](q)$ splitter sets. Finally, two nonexistence results for quasi-perfect splitter sets are presented.

cs.IT↗
arXiv2025

Linear hypermaps--modelling linear hypergraphs on surfaces

A hypergraph is linear if each pair of distinct vertices appears in at most one common edge. We say $\varGamma=(V,E)$ is an associated graph of a linear hypergraph $\mathcal{H}=(V, X)$ if for any $x\in X$, the induced subgraph $\varGamma[x]$ is a cycle, and for any $e\in E$, there exists a unique edge $y\in X$ such that $e\subseteq y$. A linear hypermap $\mathcal{M}$ is a $2$-cell embedding of a connected linear hypergraph $\mathcal{H}$'s associated graph $\varGamma$ on a compact connected surface, such that for any edge $x\in E(\mathcal{H})$, $\varGamma[x]$ is the boundary of a $2$-cell and for any $e\in E(\varGamma)$, $e$ is incident with two distinct $2$-cells. In this paper, we introduce linear hypermaps to model linear hypergraphs on surfaces and regular linear hypermaps modelling configurations on the surfaces. As an application, we classify regular linear hypermaps on the sphere and determine the total number of proper regular linear hypermaps of genus 2 to 101.

math.CO↗
arXiv2019

Directed Strongly Regular Cayley Graphs on Dihedral groups

In this paper,we construct some directed strongly regular Cayley graphs on dihedral groups,these generalizes some earlier constructions.We also characterize some certain directed strongly regular Cayley graphs on dihedral groups $D_{p^α}$,where $p$ is a prime and $α\geqslant 1$ is a positive integer.

math.CO↗
arXiv2018

On $s$-distance-transitive graphs

Distance-regular graphs have many beautiful combinatorial properties. Distance-transitive graphs have very strong symmetries, and they are distance-regular, i.e. distance-transitivity implies distance-regularity. In this paper, we give similar results, i.e. for special $s$ and graphs with other restrictions we show that $s$-distance-transitivity implies distance-regularity.

math.CO↗
arXiv2017

On the Standard Lattices

A lattice in the Euclidean space is standard if it has a basis consisting vectors whose norms equal to the length in its successive minima. In this paper, it is shown that with the $L^2$ norm all lattices of dimension $n$ are standard if and only if $n\leqslant 4$. It is also proved that with an arbitrary norm, every lattice of dimensions 1 and 2 is standard. An example of non-standard lattice of dimension $n\geqslant 3$ is given when the lattice is with the $L^1$ norm.

math.NT↗
arXiv2017

Perfect codes in circulant graphs

A perfect code in a graph $Γ= (V, E)$ is a subset $C$ of $V$ that is an independent set such that every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A total perfect code in $Γ$ is a subset $C$ of $V$ such that every vertex of $V$ is adjacent to exactly one vertex in $C$. A perfect code in the Hamming graph $H(n, q)$ agrees with a $q$-ary perfect 1-code of length $n$ in the classical setting. In this paper we give a necessary and sufficient condition for a circulant graph of degree $p-1$ to admit a perfect code, where $p$ is an odd prime. We also obtain a necessary and sufficient condition for a circulant graph of order $n$ and degree $p^l-1$ to have a perfect code, where $p$ is a prime and $p^l$ the largest power of $p$ dividing $n$. Similar results for total perfect codes are also obtained in the paper.

math.CO↗
arXiv2015

On the Solvability of 3s/nt Sum-Network---A Region Decomposition and Weak Decentralized Code Method

We study the network coding problem of sum-networks with 3 sources and n terminals (3s/nt sum-network), for an arbitrary positive integer n, and derive a sufficient and necessary condition for the solvability of a family of so-called terminal-separable sum-network. Both the condition of terminal-separable and the solvability of a terminal-separable sum-network can be decided in polynomial time. Consequently, we give another necessary and sufficient condition, which yields a faster (O(|E|) time) algorithm than that of Shenvi and Dey ([18], (O(|E|^3) time), to determine the solvability of the 3s/3t sum-network. To obtain the results, we further develop the region decomposition method in [22], [23] and generalize the decentralized coding method in [21]. Our methods provide new efficient tools for multiple source multiple sink network coding problems.

cs.IT↗
arXiv2014

Construction of Directed Strongly Regular Graphs as Generalized Cayley Graphs

Directed strongly regular graphs were introduced by Duval in 1998 as one of the possible generalization of classical strongly regular graphs to the directed case. Duval also provided several construction methods for directed strongly regular graphs. In this paper, an infinite family of directed strongly regular graphs is constructed, as generalized Cayley graphs of cyclic groups.

math.CO↗
arXiv2014

Network Coding for $3$s$/n$t Sum-Networks

A sum-network is a directed acyclic network where each source independently generates one symbol from a given field $\mathbb F$ and each terminal wants to receive the sum $($over $\mathbb F)$ of the source symbols. For sum-networks with two sources or two terminals, the solvability is characterized by the connection condition of each source-terminal pair [3]. A necessary and sufficient condition for the solvability of the $3$-source $3$-terminal $(3$s$/3$t$)$ sum-networks was given by Shenvi and Dey [6]. However, the general case of arbitrary sources/sinks is still open. In this paper, we investigate the sum-network with three sources and $n$ sinks using a region decomposition method. A sufficient and necessary condition is established for a class of $3$s$/n$t sum-networks. As a direct application of this result, a necessary and sufficient condition of solvability is obtained for the special case of $3$s$/3$t sum-networks.

cs.IT↗
arXiv2013

Encoding Complexity of Network Coding with Two Simple Multicast Sessions

The encoding complexity of network coding for single multicast networks has been intensively studied from several aspects: e.g., the time complexity, the required number of encoding links, and the required field size for a linear code solution. However, these issues as well as the solvability are less understood for networks with multiple multicast sessions. Recently, Wang and Shroff showed that the solvability of networks with two unit-rate multicast sessions (2-URMS) can be decided in polynomial time. In this paper, we prove that for the 2-URMS networks: $1)$ the solvability can be determined with time $O(|E|)$; $2)$ a solution can be constructed with time $O(|E|)$; $3)$ an optimal solution can be obtained in polynomial time; $4)$ the number of encoding links required to achieve a solution is upper-bounded by $\max\{3,2N-2\}$; and $5)$ the field size required to achieve a linear solution is upper-bounded by $\max\{2,\lfloor\sqrt{2N-7/4}+1/2\rfloor\}$, where $|E|$ is the number of links and $N$ is the number of sinks of the underlying network. Both bounds are shown to be tight.

cs.IT↗
arXiv2013

Finding normal bases over finite fields with prescribed trace self-orthogonal relations

Normal bases and self-dual normal bases over finite fields have been found to be very useful in many fast arithmetic computations. It is well-known that there exists a self-dual normal basis of $\mathbb{F}_{2^n}$ over $\mathbb{F}_2$ if and only if $4\nmid n$. In this paper, we prove there exists a normal element $α$ of $\mathbb{F}_{2^n}$ over $\mathbb{F}_{2}$ corresponding to a prescribed vector $a=(a_0,a_1,...,a_{n-1})\in \mathbb{F}_2^n$ such that $a_i={Tr}_{2^n|2}(α^{1+2^i})$ for $0\leq i\leq n-1$, where $n$ is a 2-power or odd, if and only if the given vector $a$ is symmetric ($a_i=a_{n-i}$ for all $i, 1\leq i\leq n-1$), and one of the following is true. 1) $n=2^s\geq 4$, $a_0=1$, $a_{n/2}=0$, $\sum\limits_{1\leq i\leq n/2-1, (i,2)=1}a_i=1$; 2) $n$ is odd, $(\sum\limits_{0\leq i\leq n-1}a_ix^i,x^n-1)=1$. Furthermore we give an algorithm to obtain normal elements corresponding to prescribed vectors in the above two cases. For a general positive integer $n$ with $4|n$, some necessary conditions for a vector to be the corresponding vector of a normal element of $\mathbb{F}_{2^n}$ over $\mathbb{F}_{2}$ are given. And for all $n$ with $4|n$, we prove that there exists a normal element of $\mathbb{F}_{2^n}$ over $\mathbb{F}_2$ such that the Hamming weight of its corresponding vector is 3, which is the lowest possible Hamming weight.

math.NT↗
arXiv2012

Error Correction for Cooperative Data Exchange

This paper considers the problem of error correction for a cooperative data exchange (CDE) system, where some clients are compromised or failed and send false messages. Assuming each client possesses a subset of the total messages, we analyze the error correction capability when every client is allowed to broadcast only one linearly-coded message. Our error correction capability bound determines the maximum number of clients that can be compromised or failed without jeopardizing the final decoding solution at each client. We show that deterministic, feasible linear codes exist that can achieve the derived bound. We also evaluate random linear codes, where the coding coefficients are drawn randomly, and then develop the probability for a client to withstand a certain number of compromised or failed peers and successfully deduce the complete message for any network size and any initial message distributions.

cs.IT↗
arXiv2011

Bounds on and Constructions of Unit Time-Phase Signal Sets

Digital signals are complex-valued functions on $\Z_n$. Signal sets with certain properties are required in various communication systems. Traditional signal sets consider only the time distortion during transmission. Recently, signal sets against both the time and phase distortion have been studied, and are called {\em time-phase} signal sets. Several constructions of time-phase signal sets are available in the literature. There are a number of bounds on time signal sets (also called codebooks). They are automatically bounds on time-phase signal sets, but are bad bounds. The first objective of this paper is to develop better bounds on time-phase signal sets from known bounds on time signal sets. The second objective of this paper is to construct two series of time-phase signal sets, one of which is optimal.

cs.IT↗