arXiv subjects

Chaoping Xing

Publications and source records attributed to Chaoping Xing.

At least 19 recordsLinked to original sources

arXiv2026

On Proximity Gaps for One-Point Algebraic Geometric Codes

Proximity gaps and correlated agreement underpin the soundness of interactive oracle proofs. We establish these properties for general one-point algebraic geometry (AG) codes, extending corresponding results for Reed-Solomon codes. For codes of length $n$ over $\mathbb F_q$ with designed distance $n-m$, write $ρ=m/n$ and fix $ρ\in(0,1)$. For polynomial curves of degree at most $\ell$ and any $0<\varepsilon<1$, we prove correlated agreement with multiplicative slack $\varepsilon$ and soundness $O(\ell(\ell+1)/(\varepsilon q))$ whenever $e\le(1-\varepsilon)(n-m)/2$, with no additional genus penalty. This gives asymptotically good AG code families from Garcia--Stichtenoth tower of unbounded length over a fixed field with soundness bounds independent of $n$, beyond the RS restriction $n\le q$. We also establish mutual correlated agreement up to radius $(1-\sqrtρ-η)n$, with soundness $O(\ell n/(η^3q))$ for $0<η<1-\sqrtρ$. Our half designed distance proof combines auxiliary divisors, simultaneous Berlekamp-Welch interpolation, and error-incidence counting to obtain a common polynomial representation without a Polishchuk-Spielman divisibility lemma. Finally, two counterexamples on the Garcia-Stichtenoth tower show limitations beyond half the minimum distance, at radii below the Johnson bound and with positive slack. The first requires a positive constant soundness lower bound independent of $q$ and $n$, precluding a uniform $O(1/q)$ guarantee throughout the Johnson regime. The second fixes the field and lets the relative gap above half distance tend to zero while ruling out correlated agreement for every soundness parameter $σ<1$, with positive limiting rate and relative error radius.

cs.IT↗

arXiv2026

Zassenhaus Filtrations and Golod-Shafarevich Methods for Ihara Constants

Let $X$ be a curve over $\mathbb{F}_q$. Denote by $N(X)$ and $g(X)$ the number of $\mathbb{F}_q$-rational points and the genus of $X$ respectively. The Ihara constant \begin{equation*} A(q)=\limsup_{g(X)\to\infty}\frac{N(X)}{g(X)} \end{equation*} measures the asymptotic abundance of rational points on curves over $\mathbb{F}_q$. It is a central problem in the study of algebraic geometry and has also attracted broader mathematical interest. In particular, the constant $A(2)$ is discussed in the \emph{Optimization Constants in Mathematics} collection maintained by Terence Tao et al \cite[Number 33]{DIT26}. In this paper, we study the Zassenhaus filtration of pro-$p$ groups arising from class field towers. By combining Golod--Shafarevich theorem, embedding problems and central class field theory, we establish a new criterion for the infinitude of class field towers. As an application, we obtain an improvement on the best known lower bounds of Ihara constant $A(2)$ and $A(3)$ which are explicitly given by \[ A(2)\geq 0.317779\cdots \text{ and } A(3)\geq 0.496074\cdots. \]

math.NT↗

arXiv2026

On Solvable Automorphism Groups of Function Fields over Finite Fields

The classical Hurwitz bound gives a linear upper bound in terms of the genus for automorphism groups of algebraic function fields in characteristic zero, whereas algebraic function fields in positive characteristic can have much larger automorphism groups. Usually, one considers the case where the constant field is an algebraically closed field. Here we focus on function fields over a finite constant field. It turns out that the situation is very different. We firstly investigate solvable subgroups and $p$-subgroups of $\rm{Aut}(F/\Bbb{F}_q)$, where $p=\rm{char}(\Bbb{F}_q)$ and $F/\Bbb{F}_q$ is a function field of genus $g_F\ge2$ with full constant field $\Bbb{F}_q$. We prove that there are explicit constants $C_q^{(sol)}$ and $C_q^{(p)}$, depending only on $q$, such that, for every solvable subgroup $G\le\rm{Aut}(F/\Bbb{F}_q)$ and every $p$-subgroup $P\le\rm{Aut}(F/\Bbb{F}_q)$, $$ |G|\le C_q^{(sol)}(g_F-1),\qquad |P|\le C_q^{(p)}(g_F-1).$$ In particular, one may take $C_q^{(sol)}=12q^2(q-1)$ for $q\ge4$ and $C_q^{(p)}=5q$ for every $q$. In the second part of this article, we give asymptotic lower bounds for solvable automorphism groups. For every $q$ there is an explicit constant $L_q>0$ and a sequence of function fields $F_i/\Bbb{F}_q$ with $g_{F_i}\to\infty$, together with solvable subgroups $G_i\le\rm{Aut}(F_i/\Bbb{F}_q)$, such that $$ |G_i|\ge L_q(g_{F_i}-1) $$ for all $i$. For $q\ge5$ one may take $L_q=2q(q-1)/(q-2)$. If $q=r^2$ with $r\ge3$, an unramified tower construction over the Hermitian function field gives the stronger choice $L_q=2r^3(r-1)/(r-2)$. Moreover, as an application of our tower construction above, we improve the asymptotic lower bound of Ma and Xing for subgroups of $\rm{Aut}(F/\Bbb{F}_q)$ whose order is relatively prime to $p$. As another application, we extend our method to obtain explicit linear bounds for $p$-solvable groups in terms of the genus.

math.NT↗

arXiv2026

Multi-sequences with large linear and error linear complexity from function fields

The linear complexity and the error linear complexity of multi-sequences are measures for security in stream ciphers. In this manuscript, we present a general framework for constructing periodic multi-sequences via function fields. We prove that the constructed multi-sequences possess both large linear complexity and large error linear complexity. We apply this framework of constructing multi-sequences to various maximal function fields and we obtain many new multi-sequences with various lengths and dimensions. As a byproduct, we adopt this idea and produce many new quasi-cyclic algebraic geometric codes as well.

cs.IT↗

arXiv2026

Proximity Gaps for Gabidulin Codes and Applications

Proximity gaps are central to the soundness of interactive oracle proofs of proximity (IOPPs) and polynomial commitment schemes (PCSs). An $[n,k,d]$ linear code $C\subseteq\mathbb F^n$ has a $δ$-proximity gap with error $ε$ if, for every $u_0,u_1\in\mathbb F^n$, either all points on $\ell_{u_0,u_1}=\{u_0+αu_1:α\in\mathbb F\}$ are $δ$-close to $C$, or at most an $ε$ fraction are. Although proximity gaps for Hamming-metric codes are well understood, their rank-metric counterparts remain largely unexplored despite their applications in coding theory and cryptography. In this work, we study proximity gaps for linear rank-metric codes and their cryptographic applications. First, we show that every $[n,k,d]$ linear rank-metric code $C$ over $\mathbb F_{q^m}$ admits a proximity gap for every $δ\le(d-1)/(3n)$, with error at most $q^{e+1}/q^m$, where $e=\lfloorδn\rfloor$. For Gabidulin codes, we improve the gap to $(d-1)/(2n)$ with error $10q^{n-1}/q^m$. These two proximity gaps match those for general linear Hamming-metric codes and Reed--Solomon (RS) codes, respectively. We prove the $(d-1)/(2n)$ bound is tight by constructing an infinite family of constant-rate Gabidulin codes and affine lines $\ell_{u_0,u_1}$ on which a $1-o(1)$ fraction of points are $d/(2n)$-close to the code, while $u_1$ is at least $3d/(4n)$-far from it. At the $d/(3n)$ gap, we also give a counterexample establishing a lower bound on $ε$. As applications, we construct an IOPP for interleaved Gabidulin codes by adapting the Ligero IOPP for interleaved RS codes. We then adapt the Ligero-based PCS for ordinary polynomials to obtain a $q$-linearized polynomial commitment scheme. To our knowledge, this is the first PCS framework based on rank-metric error-correcting codes.

cs.IT↗

arXiv2026

On exceptional cliques in matrix rings

We study the notion of exceptional clique, a subset of a ring such that the difference of any two distinct elements of the subset is invertible. Motivated by applications in cryptography, our main focus is to determine the largest size of an exceptional clique in the ring $Mat_{n\times n}(\mathbb{Z})$ of square $n\times n$ matrices over the integers, for every $n$. We obtain several results for the question above, both in the general case and the ``commutative'' case where we additionally require that the elements in the clique commute with each other. As highlights, we prove that, at least for some values of $n$, the largest exceptional cliques in $Mat_{n\times n}(\mathbb{Z})$ are necessarily non-commutative; we then show that for an infinite family of $n$, there are non-commutative exceptional cliques of size $n^2$, and that for every $n$ there are commutative exceptional cliques of size $\frac23 n+O(n^θ)$, for a constant $θ>\frac{11}{20}$.

math.AC↗

arXiv2026

Random Gabidulin Codes Achieve List Decoding Capacity in the Rank Metric

Gabidulin codes, serving as the rank-metric counterpart of Reed-Solomon codes, constitute an important class of maximum rank distance (MRD) codes. However, unlike the fruitful positive results about the list decoding of Reed-Solomon codes, results concerning the list decodability of Gabidulin codes in the rank metric are all negative so far. For example, in contrast to Reed-Solomon codes, which are always list decodable up to the Johnson bound in the Hamming metric, Raviv and Wachter-Zeh (IEEE TIT, 2016 and 2017) constructed a class of Gabidulin codes that are not even combinatorially list decodable beyond the unique decoding radius in the rank metric. Proving the existence of Gabidulin codes with good combinatorial list decodability in the rank metric has remained a long-standing open problem. In this paper, we resolve the aforementioned open problem by showing that, with high probability, random Gabidulin codes over sufficiently large alphabets attain the optimal generalized Singleton bound for list decoding in the rank metric. In particular, they achieve list decoding capacity in the rank metric. Our work is significantly influenced by the recent breakthroughs in the combinatorial list decodability of Reed-Solomon codes, especially the work by Brakensiek, Gopi, and Makam (STOC 2023). Our major technical contributions, which may hold independent interest, consist of the following: (1) We initiate the study of ``higher order MRD codes'' and provide a novel unified theory, which runs parallel to the theory of ``higher order MDS codes'' developed by BGM. (2) We prove a natural analog of the GM-MDS theorem, proven by Lovett (FOCS 2018) and Yildiz and Hassibi (IEEE TIT, 2019), which we call the GM-MRD theorem. In particular, our GM-MRD theorem for Gabidulin codes are strictly stronger than the GM-MDS theorem for Gabidulin codes, proven by Yildiz and Hassibi (IEEE TIT, 2019).

cs.IT↗

arXiv2025

Quasi-linear time decoding of RS and AG codes for burst errors up to the Singleton bound

Despite of tremendous research on decoding Reed-Solomon (RS) and algebraic geometry (AG) codes under the random and adversary substitution error models, few studies have explored these codes under the burst substitution error model. Burst errors are prevalent in many communication channels, such as wireless networks, magnetic recording systems, and flash memory. Compared to random and adversarial errors, burst errors often allow for the design of more efficient decoding algorithms. However, achieving both an optimal decoding radius and quasi-linear time complexity for burst error correction remains a significant challenge. The goal of this paper is to design (both list and probabilistic unique) decoding algorithms for RS and AG codes that achieve the Singleton bound for decoding radius while maintaining quasi-linear time complexity. Our idea is to build a one-to-one correspondence between AG codes (including RS codes) and interleaved RS codes with shorter code lengths (or even constant lengths). By decoding the interleaved RS codes with burst errors, we derive efficient decoding algorithms for RS and AG codes. For decoding interleaved RS codes with shorter code lengths, we can employ either the naive methods or existing algorithms. This one-to-one correspondence is constructed using the generalized fast Fourier transform (G-FFT) proposed by Li and Xing (SODA 2024). The G-FFT generalizes the divide-and-conquer technique from polynomials to algebraic function fields. More precisely speaking, assume that our AG code is defined over a function field $E$ which has a sequence of subfields $\mathbb{F}_q(x)=E_r\subseteq E_{r-1}\subseteq \cdots\subset E_1\subseteq E_0=E$ such that $E_{i-1}/E_i$ are Galois extensions for $1\le i\le r$. Then the AG code based on $E$ can be transformed into an interleaved RS code over the rational function field $\mathbb{F}_q(x)$.

cs.IT↗

arXiv2024

Coded Distributed (Batch) Matrix Multiplication over Galois Ring via RMFE

Coded Distributed Matrix Multiplication (CDMM) is a distributed matrix multiplication (DMM) for large-scale matrices through a coding scheme such that any $R$ worker node among all $N$ worker nodes can recover the final product, where $N$ corresponds to the length of the code and $R\leq N$ is called the recovery threshold. The state-of-art CDMM schemes, such as EP codes for Single DMM and GCAS codes for batch DMM, are defined over a Galois field $\mathsf{GF}(q)$ of size $q\geq N$. These are inefficient for small Galois fields such as $\mathsf{GF}(2)$ and the integer residue ring $\mathbb{Z}_{p^{e}}$ due to the lack of invertible elements for interpolation. DMM over $\mathbb{Z}_{p^{e}}$ (such as $\mathbb{Z}_{2^{64}}$ ) is well-motivated in practice due to their direct compatibility with hardware. In this work, we construct efficient CDMM over the Galois ring $\mathsf{GR}(p^e,d)$ which is an extension ring over $\mathbb{Z}_{p^{e}}$ of degree $d$, particularly, $\mathsf{GR}(p,d)=\mathsf{GF}(p^d)$ is the Galois field and $\mathsf{GR}(p^e,1)=\mathbb{Z}_{p^e}$. We first give a general CDMM framework for the batch of $n$ matrix multiplications via the famous RMFE (Cascudo et al. Crypto'18). Compared with GCSA, our construction has a smaller recovery threshold by a factor of $1/n$. Next, we optimize EP codes via batch preprocessing of the input matrices. We give two types of Single CDMM, which can achieve almost the same performance as EP codes over a Galois field with size $q\geq N$. Finally, we present the experimental analysis of our CDMM on Galois rings.

cs.DC↗

arXiv2024

New families of non-Reed-Solomon MDS codes

MDS codes have garnered significant attention due to their wide applications in practice. To date, most known MDS codes are equivalent to Reed-Solomon codes. The construction of non-Reed-Solomon (non-RS) type MDS codes has emerged as an intriguing and important problem in both coding theory and finite geometry. Although some constructions of non-RS type MDS codes have been presented in the literature, the parameters of these MDS codes remain subject to strict constraints. In this paper, we introduce a general framework of constructing $[n,k]$ MDS codes using the idea of selecting a suitable set of evaluation polynomials and a set of evaluation points such that all nonzero polynomials have at most $k-1$ zeros in the evaluation set. Moreover, these MDS codes can be proved to be non-Reed-Solomon by computing their Schur squares. Furthermore, several explicit constructions of non-RS MDS codes are given by converting to combinatorial problems. As a result, new families of non-RS MDS codes with much more flexible lengths can be obtained and most of them are not covered by the known results.

cs.IT↗

arXiv2024

Algebraic Geometry Codes for Distributed Matrix Multiplication Using Local Expansions

Code-based Distributed Matrix Multiplication (DMM) has been extensively studied in distributed computing for efficiently performing large-scale matrix multiplication using coding theoretic techniques. The communication cost and recovery threshold (i.e., the least number of successful worker nodes required to recover the product of two matrices) are two major challenges in coded DMM research. Several constructions based on Reed-Solomon (RS) codes are known, including Polynomial codes, MatDot codes, and PolyDot codes. However, these RS-based schemes are not efficient for small finite fields because the distributed order (i.e., the total number of worker nodes) is limited by the size of the underlying finite field. Algebraic geometry (AG) codes can have a code length exceeding the size of the finite field, which helps solve this problem. Some work has been done to generalize Polynomial and MatDot codes to AG codes, but the generalization of PolyDot codes to AGcodes still remains an open problem as far as we know. This is because functions of an algebraic curve do not behave as nicely as polynomials. In this work, by using local expansions of functions, we are able to generalize the three DMM schemes based on RS codes to AG codes. Specifically, we provide a construction of AG-based PolyDot codes for the first time. In addition, our AG-based Polynomial and MatDot codes achieve better recovery thresholds compared to previous AG-based DMM schemes while maintaining similar communication costs. Our constructions are based on a novel basis of the Riemann-Roch space using local expansions, which naturally generalizes the standard monomial basis of the univariate polynomial space in RS codes. In contrast, previous work used the non-gap numbers to construct a basis of the Riemann-Roch space, which can cause cancellation problems that prevent the conditions of PolyDot codes from being satisfied.

cs.IT↗

arXiv2024

A new family of binary sequences with a low correlation via elliptic curves

In the realm of modern digital communication, cryptography, and signal processing, binary sequences with a low correlation properties play a pivotal role. In the literature, considerable efforts have been dedicated to constructing good binary sequences of various lengths. As a consequence, numerous constructions of good binary sequences have been put forward. However, the majority of known constructions leverage the multiplicative cyclic group structure of finite fields $\mathbb{F}_{p^n}$, where $p$ is a prime and $n$ is a positive integer. Recently, the authors made use of the cyclic group structure of all rational places of the rational function field over the finite field $\mathbb{F}_{p^n}$, and firstly constructed good binary sequences of length $p^n+1$ via cyclotomic function fields over $\mathbb{F}_{p^n}$ for any prime $p$ \cite{HJMX24,JMX22}. This approach has paved a new way for constructing good binary sequences. Motivated by the above constructions, we exploit the cyclic group structure on rational points of elliptic curves to design a family of binary sequences of length $2^n+1+t$ with a low correlation for many given integers $|t|\le 2^{(n+2)/2}$. Specifically, for any positive integer $d$ with $\gcd(d,2^n+1+t)=1$, we introduce a novel family of binary sequences of length $2^n+1+t$, size $q^{d-1}-1$, correlation bounded by $(2d+1) \cdot 2^{(n+2)/2}+ |t|$, and a large linear complexity via elliptic curves.

math.NT↗

arXiv2024

Encoding of algebraic geometry codes with quasi-linear complexity $O(N\log N)$

Fast encoding and decoding of codes have been always an important topic in code theory as well as complexity theory. Although encoding is easier than decoding in general, designing an encoding algorithm of codes of length $N$ with quasi-linear complexity $O(N\log N)$ is not an easy task. Despite the fact that algebraic geometry codes were discovered in the early of 1980s, encoding algorithms of algebraic geometry codes with quasi-linear complexity $O(N\log N)$ have not been found except for the simplest algebraic geometry codes--Reed-Solomon codes. The best-known encoding algorithm of algebraic geometry codes based on a class of plane curves has quasi-linear complexity at least $O(N\log^2 N)$. In this paper, we design an encoding algorithm of algebraic geometry codes with quasi-linear complexity $O(N\log N)$. Our algorithm works well for a large class of algebraic geometry codes based on both plane and non-plane curves. The main idea of this paper is to generalize the divide-and-conquer method from the fast Fourier Transform over finite fields to algebraic curves. Suppose we consider encoding of algebraic geometry codes based on an algebraic curve ${\mathcal X}$ over $\mathbb{F}_q$. We first consider a tower of Galois coverings ${\mathcal X}={\mathcal X}_0\rightarrow{\mathcal X}_1\rightarrow\cdots\rightarrow{\mathcal X}_r$ over a finite field $\mathbb{F}_q$, i.e., their function field tower $\mathbb{F}_q({\mathcal X}_0)\supsetneq\mathbb{F}_q({\mathcal X}_{1})\supsetneq\cdots \supsetneq\mathbb{F}_q({\mathcal X}_r)$ satisfies that each of extension $\mathbb{F}_q({\mathcal X}_{i-1})/\mathbb{F}_q({\mathcal X}_i)$ is a Galois extension and the extension degree $[\mathbb{F}_q({\mathcal X}_{i-1}):\mathbb{F}_q({\mathcal X}_i)]$ {is a constant}. Then encoding of an algebraic geometry code based on ${\mathcal X}$ is reduced to the encoding of an algebraic geometry code based on ${\mathcal X}_r$.

cs.CC↗

arXiv2024

Asymptotic construction of locally repairable codes with multiple recovering sets

Locally repairable codes have been extensively investigated due to practical applications in distributed and cloud storage systems in recent years. However, not much work on asymptotic behavior of locally repairable codes has been done. In particular, there is few result on constructive lower bound of asymptotic behavior of locally repairable codes with multiple recovering sets. In this paper, we construct some families of asymptotically good locally repairable codes with multiple recovering sets via automorphism groups of function fields of the Garcia-Stichtenoth towers. The main advantage of our construction is to allow more flexibility of localities.

cs.IT↗

arXiv2023

Fast Fourier transform via automorphism groups of rational function fields

The Fast Fourier Transform (FFT) over a finite field $\mathbb{F}_q$ computes evaluations of a given polynomial of degree less than $n$ at a specifically chosen set of $n$ distinct evaluation points in $\mathbb{F}_q$. If $q$ or $q-1$ is a smooth number, then the divide-and-conquer approach leads to the fastest known FFT algorithms. Depending on the type of group that the set of evaluation points forms, these algorithms are classified as multiplicative (Math of Comp. 1965) and additive (FOCS 2014) FFT algorithms. In this work, we provide a unified framework for FFT algorithms that include both multiplicative and additive FFT algorithms as special cases, and beyond: our framework also works when $q+1$ is smooth, while all known results require $q$ or $q-1$ to be smooth. For the new case where $q+1$ is smooth (this new case was not considered before in literature as far as we know), we show that if $n$ is a divisor of $q+1$ that is $B$-smooth for a real $B>0$, then our FFT needs $O(Bn\log n)$ arithmetic operations in $\mathbb{F}_q$. Our unified framework is a natural consequence of introducing the algebraic function fields into the study of FFT.

cs.CC↗

arXiv2023

Nonlinear Codes with Low Redundancy

Determining the largest size, or equivalently finding the lowest redundancy, of q-ary codes for given length and minimum distance is one of the central and fundamental problems in coding theory. Inspired by the construction of Varshamov-Tenengolts (VT for short) codes via check-sums, we provide an explicit construction of nonlinear codes with lower redundancy than linear codes under the same length and minimum distance. Similar to the VT codes, our construction works well for small distance (or even constant distance). Furthermore, we design quasi-linear time decoding algorithms for both erasure and adversary errors.

cs.IT↗

arXiv2023

A Lower Bound on the List-Decodability of Insdel Codes

For codes equipped with metrics such as Hamming metric, symbol pair metric or cover metric, the Johnson bound guarantees list-decodability of such codes. That is, the Johnson bound provides a lower bound on the list-decoding radius of a code in terms of its relative minimum distance $δ$, list size $L$ and the alphabet size $q.$ For study of list-decodability of codes with insertion and deletion errors (we call such codes insdel codes), it is natural to ask the open problem whether there is also a Johnson-type bound. The problem was first investigated by Wachter-Zeh and the result was amended by Hayashi and Yasunaga where a lower bound on the list-decodability for insdel codes was derived. The main purpose of this paper is to move a step further towards solving the above open problem. In this work, we provide a new lower bound for the list-decodability of an insdel code. As a consequence, we show that unlike the Johnson bound for codes under other metrics that is tight, the bound on list-decodability of insdel codes given by Hayashi and Yasunaga is not tight. Our main idea is to show that if an insdel code with a given Levenshtein distance $d$ is not list-decodable with list size $L$, then the list decoding radius is lower bounded by a bound involving $L$ and $d$. In other words, if the list decoding radius is less than this lower bound, the code must be list-decodable with list size $L$. At the end of the paper we use such bound to provide an insdel-list-decodability bound for various well-known codes, which has not been extensively studied before.

cs.IT↗

arXiv2023

Codes with Biochemical Constraints and Single Error Correction for DNA-Based Data Storage

In DNA-based data storage, DNA codes with biochemical constraints and error correction are designed to protect data reliability. Single-stranded DNA sequences with secondary structure avoidance (SSA) help to avoid undesirable secondary structures which may cause chemical inactivity. Homopolymer run-length limit and GC-balanced limit also help to reduce the error probability of DNA sequences during synthesizing and sequencing. In this letter, based on a recent work \cite{bib7}, we construct DNA codes free of secondary structures of stem length $\geq m$ and have homopolymer run-length $\leq\ell$ for odd $m\leq11$ and $\ell\geq3$ with rate $1+\log_2ρ_m-3/(2^{\ell-1}+\ell+1)$, where $ρ_m$ is in Table \ref{tm}. In particular, when $m=3$, $\ell=4$, its rate tends to 1.3206 bits/nt, beating a previous work by Benerjee {\it et al.}. We also construct DNA codes with all of the above three constraints as well as single error correction. At last, codes with GC-locally balanced constraint are presented.

cs.IT↗