arXiv subjects

Liming Ma

Publications and source records attributed to Liming Ma.

At least 19 recordsLinked to original sources

arXiv2025

The automorphism groups of a maximal function field with the second largest genus and its algebraic geometry codes

In this manuscript, we investigate the automorphism group of a maximal function field with the second largest possible genus over finite field of even characteristic, which is called the Abd\'on--Torres function field. As an application, we determine the automorphism groups of one-point algebraic geometry codes from such a maximal function field. It turns out that the automorphism groups of one-point algebraic geometry codes agree with that of the Abd\'on--Torres function field except for the trivial cases. Moreover, we provide a family of maximal function fields with explicit defining equations via considering fixed subfields with respect to some subgroups of automorphism group of the Abd\'on--Torres function field.

math.NT↗

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 $\mathbb{F}_q(x)$.

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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.

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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.

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arXiv2022

Binary sequences with a low correlation via cyclotomic function fields with odd characteristics

Sequences with a low correlation have very important applications in communications, cryptography, and compressed sensing. In the literature, many efforts have been made to construct good sequences with various lengths where binary sequences attracts great attention. As a result, various constructions of good binary sequences have been proposed. However, most of the known constructions made use of the multiplicative cyclic group structure of finite field $\mathbb{F}_{p^n}$ for a prime $p$ and a positive integer $n$. In fact, all $p^n+1$ rational places including the place at infinity of the rational function field over $\mathbb{F}_{p^n}$ form a cyclic structure under an automorphism of order $p^n+1$. In this paper, we make use of this cyclic structure to provide an explicit construction of binary sequences with a low correlation of length $p^n+1$ via cyclotomic function fields over $\mathbb{F}_{p^n}$ for any odd prime $p$. Each family of binary sequences has size $p^n-2$ and its correlation is upper bounded by $4+\lfloor 2\cdot p^{n/2}\rfloor$. To the best of our knowledge, this is the first construction of binary sequences with a low correlation of length $p^n+1$ for odd prime $p$. Moreover, our sequences can be constructed explicitly and have competitive parameters.

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arXiv2022

A new construction of nonlinear codes via algebraic function fields

In coding theory, constructing codes with good parameters is one of the most important and fundamental problems. Though a great many of good codes have been produced, most of them are defined over alphabets of sizes equal to prime powers. In this paper, we provide a new explicit construction of $(q+1)$-ary nonlinear codes via algebraic function fields, where $q$ is a prime power. Our codes are constructed by evaluations of rational functions at all rational places of the algebraic function field. Compared with algebraic geometry codes, the main difference is that we allow rational functions to be evaluated at pole places. After evaluating rational functions from a union of Riemann-Roch spaces, we obtain a family of nonlinear codes over the alphabet $\mathbb{F}_{q}\cup \{\infty\}$. It turns out that our codes have better parameters than those obtained from MDS codes or good algebraic geometry codes via code alphabet extension and restriction.

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arXiv2022

Good locally repairable codes via propagation rules

In classical coding theory, it is common to construct new codes via propagation rules. There are various propagation rules to construct classical block codes. However, propagation rules have not been extensively explored for constructions of locally repairable codes. In this paper, we introduce a few propagation rules to construct good locally repairable codes. To our surprise, these simple propagation rules produce a few interesting results. Firstly, by concatenating a locally repairable code as an inner code with a classical block code as an outer code, we obtain quite a few dimension-optimal binary locally repairable codes. Secondly, from this concatenation, we explicitly build a family of locally repairable codes that exceeds the Zyablov-type bound. Thirdly, by a lengthening propagation rule that adds some rows and columns from a parity-check matrix of a given linear code, we are able to produce a family of dimension-optimal binary locally repairable codes from the extended Hamming codes, and to convert a classical maximum distance separable (MDS) code into a Singleton-optimal locally repairable code. Furthermore, via the lengthening propagation rule, we greatly simplify the construction of a family of locally repairable codes in \cite[Theorem 5]{MX20} that breaks the asymptotic Gilbert-Varshamov bound. In addition, we make use of three other propagation rules to produce more dimension-optimal binary locally repairable codes. Finally, one of phenomena that we observe in this paper is that some trivial propagation rules in classical block codes do not hold anymore for locally repairable codes.

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arXiv2021

Binary sequences with low correlation via cyclotomic function fields

Due to wide applications of binary sequences with low correlation to communications, various constructions of such sequences have been proposed in literature. However, most of the known constructions via finite fields make use of the multiplicative cyclic group of $\F_{2^n}$. It is often overlooked in this community that all $2^n+1$ rational places (including "place at infinity") of the rational function field over $\F_{2^n}$ form a cyclic structure under an automorphism of order $2^n+1$. In this paper, we make use of this cyclic structure to provide an explicit construction of families of binary sequences of length $2^n+1$ via the finite field $\F_{2^n}$. Each family of sequences has size $2^n-1$ and its correlation is upper bounded by $\lfloor 2^{(n+2)/2}\rfloor$. Our sequences can be constructed explicitly and have competitive parameters. In particular, compared with the Gold sequences of length $2^n-1$ for even $n$, we have larger length and smaller correlation although the family size of our sequences is slightly smaller.

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arXiv2021

An improvement of the Hasse-Weil bound for Artin-Schreier curves via cyclotomic function fields

The corresponding Hasse-Weil bound was a major breakthrough in history of mathematics. It has found many applications in mathematics, coding theory and theoretical computer science. In general, the Hasse-Weil bound is tight and cannot be improved. However, the Hasse-Weil bound is no longer tight when it is applied to some specific classes of curves. One of the examples where the Hasse-Weil bound is not tight is the family of Artin-Schreier curves. Due to various applications of Artin-Schreier curves to coding, cryptography and theoretical computer science, researchers have made great effort to improve the Hasse-Weil bound for Artin-Schreier curves. In this paper, we focus on the number of rational places of the Artin-Schreier curve defined by $y^p-y=f(x)$ over the finite field $\mathbb{F}_q$ of characteristic $p$, where $f(x)$ is a polynomial in $\mathbb{F}_q[x]$. Our road map for attacking this problem works as follows. We first show that the function field $E_f:=\mathbb{F}_q(x,y)$ of the Artin-Schreier curve $y^p-y=f(x)$ is a subfield of some cyclotomic function field. We then make use of the class field theory to prove that the number of points of the curve is upper bounded by a function of a minimum distance of a linear code. By analyzing the minimum distance of this linear code, we can improve the Hasse-Weil bound and Serre bound for Artin-Schreier curves.

math.NT↗

arXiv2020

The group structures of automorphism groups of elliptic function fields over finite fields and their applications to optimal locally repairable codes

The automorphism group of an elliptic curve over an algebraically closed field is well known. However, for various applications in coding theory and cryptography, we usually need to apply automorphisms defined over a finite field. Although we believe that the automorphism group of an elliptic curve over a finite field is well known in the community, we could not find this in the literature. Nevertheless, in this paper we show the group structure of the automorphism group of an elliptic curve over a finite field. More importantly, we characterize subgroups and abelian subgroups of the automorphism group of an elliptic curve over a finite field. Despite of theoretical interest on this topic, our research is largely motivated by constructions of optimal locally repairable codes. The first research to make use of automorphism group of function fields to construct optimal locally repairable codes was given in a paper \cite{JMX20} where automorphism group of a projective line was employed. The idea was further generated to an elliptic curve in \cite{MX19} where only automorphisms fixing the point at infinity were used. Because there are at most $24$ automorphisms of an elliptic curve fixing the point at infinity, the locality of optimal locally repairable codes from this construction is upper bounded by $23$. One of the main motivation to study subgroups and abelian subgroups of the automorphism group of an elliptic curve over a finite field is to remove the constraints on locality.

math.NT↗

arXiv2019

A new construction of nonlinear codes via rational function fields

It is well known that constructing codes with good parameters is one of the most important and fundamental problems in coding theory. Though a great many of good codes have been produced, most of them are defined over alphabets of sizes equal to prime powers. In this paper, we provide a new explicit construction of (q+1)-ary nonlinear codes via rational function fields, where q is a prime power. Our codes are constructed by evaluations of rational functions at all the rational places (including the place of "infinity") of the rational function field. Compared to the rational algebraic geometry codes, the main difference is that we allow rational functions to be evaluated at pole places. After evaluating rational functions from a union of Riemann-Roch spaces, we obtain a family of nonlinear codes with length q+1 over the alphabet $\mathbb{F}_{q}\cup \{\infty\}$. As a result, our codes have reasonable parameters as they are very close to the Singleton bound. Furthermore, our codes have better parameters than those obtained from MDS codes via code alphabet restriction or extension.

cs.IT↗

arXiv2019

Constructive asymptotic bounds of locally repairable codes via function fields

Locally repairable codes have been investigated extensively in recent years due to practical applications in distributed and cloud storage systems. However, there are few asymptotical constructions of locally repairable codes in the literature. In this paper, we provide an explicit asymptotic construction of locally repairable codes over arbitrary finite fields from local expansions of functions at a rational place. This construction gives a Tsfasman-Vladut-Zink type bound for locally repairable codes. Its main advantage is that there are no constraints on both locality and alphabet size. Furthermore, we show that the Gilbert-Varshamov type bound on locally repairable codes over non-prime finite fields can be exceeded for sufficiently large alphabet size.

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arXiv2017

Optimal locally repairable codes via elliptic curves

Constructing locally repairable codes achieving Singleton-type bound (we call them optimal codes in this paper) is a challenging task and has attracted great attention in the last few years. Tamo and Barg \cite{TB14} first gave a breakthrough result in this topic by cleverly considering subcodes of Reed-Solomon codes. Thus, $q$-ary optimal locally repairable codes from subcodes of Reed-Solomon codes given in \cite{TB14} have length upper bounded by $q$. Recently, it was shown through extension of construction in \cite{TB14} that length of $q$-ary optimal locally repairable codes can be $q+1$ in \cite{JMX17}. Surprisingly it was shown in \cite{BHHMV16} that, unlike classical MDS codes, $q$-ary optimal locally repairable codes could have length bigger than $q+1$. Thus, it becomes an interesting and challenging problem to construct $q$-ary optimal locally repairable codes of length bigger than $q+1$. In the present paper, we make use of rich algebraic structures of elliptic curves to construct a family of $q$-ary optimal locally repairable codes of length up to $q+2\sqrt{q}$. It turns out that locality of our codes can be as big as $23$ and distance can be linear in length.

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arXiv2017

Construction of asymptotically good locally repairable codes via automorphism groups of function fields

Locally repairable codes have been investigated extensively in recent years due to practical application in distributed storage as well as theoretical interest. However, not much work on asymptotical behavior of locally repairable codes has been done until now. In particular, there is a little result on constructive lower bound on asymptotical behavior of locally repairable codes. In this paper, we extend the construction given in \cite{BTV17} via automorphism groups of function field towers. The main advantage of our construction is to allow more flexibility of locality. Furthermore, we show that the Gilbert-Varshamov type bound on locally repairable codes can be improved for all sufficiently large alphabet size $q$.

cs.IT↗

arXiv2017

Construction of optimal locally repairable codes via automorphism groups of rational function fields

Locally repairable codes, or locally recoverable codes (LRC for short) are designed for application in distributed and cloud storage systems. Similar to classical block codes, there is an important bound called the Singleton-type bound for locally repairable codes. In this paper, an optimal locally repairable code refers to a block code achieving this Singleton-type bound. Like classical MDS codes, optimal locally repairable codes carry some very nice combinatorial structures. Since introduction of the Singleton-type bound for locally repairable codes, people have put tremendous effort on constructions of optimal locally repairable codes. Due to hardness of this problem, there are few constructions of optimal locally repairable codes in literature. Most of these constructions are realized via either combinatorial or algebraic structures. In this paper, we employ automorphism groups of rational function fields to construct optimal locally repairable codes by considering the group action on the projective lines over finite fields. It turns out that we are able to construct optimal locally repairable codes with reflexibility of locality as well as smaller alphabet size comparable to the code length. In particular, we produce new families of $q$-ary locally repairable codes, including codes of length $q+1$ via cyclic groups and codes via dihedral groups.

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arXiv2017

On subfields of the Hermitian function fields involving the involution automorphism

A function field over a finite field is called maximal if it achieves the Hasse-Weil bound. Finding possible genera that maximal function fields achieve has both theoretical interest and practical applications to coding theory and other topics. As a subfield of a maximal function field is also maximal, one way to find maximal function fields is to find all subfields of a maximal function field. Due to the large automorphism group of the Hermitian function field, it is natural to find as many subfields of the Hermitian function field as possible. In literature, most of papers studied subfields fixed by subgroups of the decomposition group at one point (usually the point at infinity). This is because it becomes much more complicated to study the subfield fixed by a subgroup that is not contained in the decomposition group at one point. In this paper, we study subfields of the Hermitian function field fixed by subgroups that are not contained in the decomposition group of any point except the cyclic subgroups. It turns out that some new maximal function fields are found.

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