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Keqin Feng

Publications and source records attributed to Keqin Feng.

At least 19 recordsLinked to original sources
arXiv2026

Constructions of $q$-ary Golay Complementary Pairs Over Flexible Non-Power-of-Two Lengths

Golay complementary pair (GCP), first introduced by Golay in 1951, has been extensively studied and widely applied in communication systems. A $q$-ary GCP $\{\mathbf{A},\mathbf{B}\}$ consists of two $q$-ary complex sequences $\mathbf{A}=(A_0,\cdots,A_{M-1})$ and $\mathbf{B}=({B}_0,\cdots,{B}_{M-1})$ of equal length $M$, where $\textit{A}_i,\textit{B}_i\in\{\xi^a:0\leq a\leq q-1\}$ with $\xi=e^{\frac{2\pi\sqrt{-1}}{q}}$.In this paper,we prove that the existence of a quaternary ($q=4$) GCP of length $M$ is equivalent to the explicit constructibility of ($4h$)-ary GCPs of length $2^mM$ for all integers $h,m\geq1$. All proposed sequences are constructed via extended Boolean functions (EBFs), and the direct construction yields GCPs with more flexible length ranges than all previous relevant results.

cs.IT↗
arXiv2025

Asymptotically Optimal Aperiodic Doppler Resilient Complementary Sequence Sets Via Generalized Quasi-Florentine Rectangles

Doppler-resilient complementary sequence (DRCS) sets play a vital role in modern communication and sensing systems, particularly in high-mobility environments. This work makes two primary contributions. First, we refine the definition of quasi-Florentine rectangles to a more general form,termed generalized quasi-Florentine rectangles, and propose a systematic method for their construction. Second, we propose several sets of aperiodic DRCS based on generalized quasi Florentine rectangles and Butson-type Hadamard matrices. The proposed aperiodic DRCS sets are shown to be asymptotically optimal with respect to the lower bound of aperiodic DRCS sets.

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arXiv2025

On the quantum chromatic number of Hamming and generalized Hadamard graphs

As a fundamental metric for quantifying quantum advantage in non-local games, the quantum chromatic number reveals the power of entanglement in distributed tasks. In this paper, we investigate this parameter for $q$-ary Hamming graphs and a generalization of Hadamard graphs. Our main results establish an exponential separation between the quantum and classical chromatic numbers for both graph families, and determine the exact quantum chromatic numbers in several regimes. Our analysis builds on known upper and lower bounds via modulus-one orthogonal representations and minimum eigenvalues, respectively. Previous results for Hamming graphs $H(n,q,d)$ were restricted to specific cases: the minimum eigenvalue was only identified for $d > (q-1)n/q$, while modulus-one orthogonal representations had only been constructed for the binary case ($q=2$) with $d \ge n/2$. In this work, we fill several gaps in the existing literature by developing a linear programming approach to construct modulus-one orthogonal representations for arbitrary relative distances, and using the trace method to determine the minimum eigenvalues in the regime where $d$ lies slightly below the threshold $(q-1)n/q$. For generalized Hadamard graphs over cyclic groups and finite fields, by determining their minimum eigenvalues, we show that the spectral lower bound matches the natural upper bound on the quantum chromatic number. On the classical side, we apply the method of forbidden intersection pattern of Frankl and R\"odl to obtain an exponential lower bound on the chromatic number, thereby quantifying the separation between the quantum and classical quantities.

math.CO↗
arXiv2025

New Constructions of Locally Perfect Nonlinear Functions and Their Application to Sequence Sets With Low Ambiguity Zone

Low Ambiguity Zone (LAZ) sequences play a pivotal role in modern integrated sensing and communication (ISAC) systems. Recently, Wang \textit{et al.} [arXiv:2501.11313] proposed a definition of locally perfect nonlinear functions (LPNFs) and constructed three classes of both periodic and aperiodic LAZ sequence sets with flexible parameters by applying such functions and interleaving techniques. Some of these LAZ sequence sets are asymptotically optimal with respect to the Ye-Zhou-Fan-Liu-Lei-Tang bounds under certain conditions. In this paper, we present constructions of three new classes of LPNFs with new parameters. Based on these LPNFs, we further propose a series of LAZ sequence sets that offer more flexible parameters. Furthermore, our results show that some of these classes are asymptotically optimal in both the periodic and aperiodic cases, respectively.

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arXiv2024

Quantum chromatic numbers of some graphs in Hamming schemes

The study of quantum chromatic numbers of graphs is a hot research topic in recent years. However, the infinite family of graphs with known quantum chromatic numbers are rare, as far as we know, the only known such graphs (except for complete graphs, cycles, bipartite graphs and some trivial cases) are the Hadamard graphs $H_n$ with $2^n$ vertices and $n$ a multiple of $4$. In this paper, we consider the graphs in Hamming schemes, we determined the quantum chromatic numbers of one class of such graphs. Notably, this is the second known family of graphs whose quantum chromatic numbers are explicitly determined except for some cases aforementioned. We also provide some bounds for the quantum chromatic numbers of some other graphs in Hamming schemes. Consequently, we can obtain the quantum chromatic numbers of products of some graphs.

math.CO↗
arXiv2024

Quadratic twists of tiling number elliptic curves

A positive integer $n$ is called a tiling number if the equilateral triangle can be dissected into $nk^2$ congruent triangles for some integer $k$. An integer $n>3$ is tiling number if and only if at least one of the elliptic curves $E^{(\pm n)}:\pm ny^2=x(x-1)(x+3)$ has positive Mordell-Weil rank. Let $A$ denote one of the two curves. In this paper, using Waldspurger formula and an induction method, for $n\equiv 3,7\mod 24$ positive square-free, as well as some other residue classes, we express the parity of analytic Sha of $A$ in terms of the genus number $g(m):=\#2\mathrm{Cl}(\mathbb{Q}(\sqrt{-m}))$ as $m$ runs over factors of $n$. Together with $2$-descent method which express $\mathrm{dim}_{\mathbb{F}_2}\mathrm{Sel}_2(A/\mathbb{Q})/A[2]$ in terms of the corank of a matrix of $\mathbb{F}_2$-coefficients, we show that for $n\equiv 3,7\mod 24$ positive square-free, the analytic Sha of $A$ being odd is equivalent to that $\mathrm{Sel}_2(A/\mathbb{Q})/A[2]$ being trivial, as predicted by the BSD conjecture. We also show that, among the residue classes $3$, resp. $7\mod 24$, the subset of $n$ such that both of $E^{(n)}$ and $E^{(-n)}$ have analytic Sha odd is of limit density $0.288\cdots$ and $0.144\cdots$, respectively, in particular, they are non-tiling numbers. This exhibits two new phenomena on tiling number elliptic curves: firstly, the limit density is different from the general phenomenon on elliptic curves predicted by Bhargava-Kane-Lenstra-Poonen-Rains; secondly, the joint distribution has different behavior among different residue classes.

math.NT↗
arXiv2023

The Weight Distributions of Two Classes of Linear Codes From Perfect Nonlinear Functions

In this paper, we employ general results on the value distributions of perfect nonlinear functions from $\mathbb{F}_{p^m}$ to $\mathbb{F}_p$ together with a specific group action to give a unified approach to determining the weight distributions of two classes of linear codes over $\mathbb{F}_p$ constructed from perfect nonlinear functions, where $p$ is an odd prime number and $m\in\mathbb{N}_+$.

cs.IT↗
arXiv2023

Circular External Difference Families: Construction and Non-Existence

The circular external difference family and its strong version, which themselves are of independent combinatorial interest, were proposed as variants of the difference family to construct new unconditionally secure non-malleable threshold schemes. In this paper, we present new results regarding the construction and non-existence of (strong) circular external difference families, thereby solving several open problems on this topic.

math.CO↗
arXiv2023

Constructions of $k$-uniform states in heterogeneous systems

A pure quantum state of $n$ parties associated with the Hilbert space $\CC^{d_1}\otimes \CC^{d_2}\otimes\cdots\otimes \CC^{d_n}$ is called $k$-uniform if all the reductions to $k$-parties are maximally mixed. The $n$ partite system is called homogenous if the local dimension $d_1=d_2=\cdots=d_n$, while it is called heterogeneous if the local dimension are not all equal. $k$-uniform sates play an important role in quantum information theory. There are many progress in characterizing and constructing $k$-uniform states in homogeneous systems. However, the study of entanglement for heterogeneous systems is much more challenging than that for the homogeneous case. There are very few results known for the $k$-uniform states in heterogeneous systems for $k>3$. We present two general methods to construct $k$-uniform states in the heterogeneous systems for general $k$. The first construction is derived from the error correcting codes by establishing a connection between irredundant mixed orthogonal arrays and error correcting codes. We can produce many new $k$-uniform states such that the local dimension of each subsystem can be a prime power. The second construction is derived from a matrix $H$ meeting the condition that $H_{A\times \bar{A}}+H^T_{\bar{A}\times A}$ has full rank for any row index set $A$ of size $k$. These matrix construction can provide more flexible choices for the local dimensions, i.e., the local dimensions can be any integer (not necessarily prime power) subject to some constraints. Our constructions imply that for any positive integer $k$, one can construct $k$-uniform states of a heterogeneous system in many different Hilbert spaces.

quant-ph↗
arXiv2022

Arithmetic autocorrelation distribution of binary $m$-sequences

Binary $m$-sequences are ones with the largest period $n=2^m-1$ among the binary sequences produced by linear shift registers with length $m$. They have a wide range of applications in communication since they have several desirable pseudorandomness such as balance, uniform pattern distribution and ideal (classical) autocorrelation. In his reseach on arithmetic codes, Mandelbaum \cite{9Mand} introduces a 2-adic version of classical autocorrelation of binary sequences, called arithmetic autocorrelation. Later, Goresky and Klapper \cite{3G1,4G2,5G3,6G4} generalize this notion to nonbinary case and develop several properties of arithmetic autocorrelation related to linear shift registers with carry. Recently, Z. Chen et al. \cite{1C1} show an upper bound on arithmetic autocorrelation of binary $m$-sequences and raise a conjecture on absolute value distribution on arithmetic autocorrelation of binary $m$-sequences.

cs.IT↗
arXiv2022

The 4-Adic Complexity of Interleaved Quaternary Sequences of Even Length with Optimal Autocorrelation

Su et al. proposed several new classes of quaternary sequences of even length with optimal autocorrelation interleaved by twin-prime sequences pairs, GMW sequences pairs or binary cyclotomic sequences of order four in \cite{S1}. In this paper, we determine the 4-adic complexity of these quaternary sequences with period $2n$ by using correlation function and the "Gauss periods" of order four and "quadratic Gauss sums" on finite field $\mathbb{F}_n$ and valued in $\mathbb{Z}^{*}_{4^{2n}-1}$. Our results show that they are safe enough to resist the attack of the rational approximation algorithm.

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arXiv2021

Optimal Combinatorial Neural Codes with Matched Metric $δ_{r}$: Characterization and Constructions

Based on the theoretical neuroscience, G. Cotardo and A. Ravagnavi in \cite{CR} introduced a kind of asymmetric binary codes called combinatorial neural codes (CN codes for short), with a "matched metric" $δ_{r}$ called asymmetric discrepancy, instead of the Hamming distance $d_{H}$ for usual error-correcting codes. They also presented the Hamming, Singleton and Plotkin bounds for CN codes with respect to $δ_{r}$ and asked how to construct the CN codes $\cC$ with large size $|\cC|$ and $δ_{r}(\cC).$ In this paper we firstly show that a binary code $\cC$ reaches one of the above bounds for $δ_{r}(\cC)$ if and only if $\cC$ reaches the corresponding bounds for $d_H$ and $r$ is sufficiently closed to 1. This means that all optimal CN codes come from the usual optimal codes. %(perfect codes, MDS codes or the codes meet the usual Plotkin bound). Secondly we present several constructions of CN codes with nice and flexible parameters $(n,K, δ_r(\cC))$ by using bent functions.

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arXiv2021

Constructions on Real Approximate Mutually Unbiased Bases

Mutually unbiased bases (MUB) have many applications in quantum information processing and quantum cryptography. Several complex MUB's in $\mathbb{C}^d$ for some dimension $d$ and with larger size have been constructed. On the other hand, real MUB's with larger size are rare which lead to consider constructing approximate MUB (AMUB). In this paper we present a general and useful way to get real AMUB in $\mathbb{R}^{2d}$ from any complex AMUB in $\mathbb{C}^d$. From this method we present many new series of real AMUB's with parameters better than previous results.

quant-ph↗
arXiv2021

4-Adic Complexity of Interleaved Quaternary Sequences

Tang and Ding \cite{X. Tang} present a series of quaternary sequences $w(a, b)$ interleaved by two binary sequences $a$ and $b$ with ideal autocorrelation and show that such interleaved quaternary sequences have optimal autocorrelation. In this paper we consider the 4-adic complexity $FC_{w}(4)$ of such quaternary sequence $w=w(a, b)$. We present a general formula on $FC_{w}(4)$, $w=w(a, b)$. As a direct consequence, we obtain a general lower bound $FC_{w}(4)\geq\log_{4}(4^{n}-1)$ where $2n$ is the period of the sequence $w$. By taking $a$ and $b$ to be several types of known binary sequences with ideal autocorrelation ($m$-sequences, twin-prime, Legendre, Hall sequences and their complement, shift or sample sequences), we compute the exact values of $FC_{w}(4)$, $w=w(a, b)$ and show that in most cases $FC_{w}(4)$ reaches or nearly reaches the maximum value $\log_{4}(4^{2n}-1)$. Our results show that the 4-adic complexity of the quaternary sequences defined in \cite{X. Tang} are large enough to resist the attack of the rational approximation algorithm.

cs.IT↗
arXiv2021

On the 4-Adic Complexity of Quaternary Sequences with Ideal Autocorrelation

In this paper, we determine the 4-adic complexity of the balanced quaternary sequences of period $2p$ and $2(2^n-1)$ with ideal autocorrelation defined by Kim et al. (ISIT, pp. 282-285, 2009) and Jang et al. (ISIT, pp. 278-281, 2009), respectively. Our results show that the 4-adic complexity of the quaternary sequences defined in these two papers is large enough to resist the attack of the rational approximation algorithm.

cs.IT↗
arXiv2021

Linear Complexity of Binary Interleaved Sequences of Period 4n

Binary periodic sequences with good autocorrelation property have many applications in many aspects of communication. In past decades many series of such binary sequences have been constructed. In the application of cryptography, such binary sequences are required to have larger linear complexity. Tang and Ding \cite{X. Tang} presented a method to construct a series of binary sequences with period 4$n$ having optimal autocorrelation. Such sequences are interleaved by two arbitrary binary sequences with period $n\equiv 3\pmod 4$ and ideal autocorrelation. In this paper we present a general formula on the linear complexity of such interleaved sequences. Particularly, we show that the linear complexity of such sequences with period 4$n$ is not bigger than $2n+2$. Interleaving by several types of known binary sequences with ideal autocorrelation ($m$-sequences, Legendre, twin-prime and Hall's sequences), we present many series of such sequences having the maximum value $2n+2$ of linear complexity which gives an answer of a problem raised by N. Li and X. Tang \cite{N. Li}. Finally, in the conclusion section we show that it can be seen easily that the 2-adic complexity of all such interleaved sequences reaches the maximum value $\log_{2}(2^{4n}-1)$.

cs.IT↗
arXiv2021

Determination of the Autocorrelation Distribution and 2-Adic Complexity of Generalized Cyclotomic Binary Sequences of Order 2 with Period pq

The generalized cyclotomic binary sequences $S=S(a, b, c)$ with period $n=pq$ have good autocorrelation property where $(a, b, c)\in \{0, 1\}^3$ and $p, q$ are distinct odd primes. For some cases, the sequences $S$ have ideal or optimal autocorrelation. In this paper we determine the autocorrelation distribution and 2-adic complexity of the sequences $S=S(a, b, c)$ for all $(a, b, c)\in \{0, 1\}^3$ in a unified way by using group ring language and a version of quadratic Gauss sums valued in group ring $R=\mathbb{Z}[Γ]$ where $Γ$ is a cyclic group of order $n$.

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