arXiv subjects

Chaoping Xing

Publications and source records attributed to Chaoping Xing.

At least 73 records · Page 4Linked to original sources

arXiv2017

The asymptotic behavior of automorphism groups of function fields over finite fields

The purpose of this paper is to investigate the asymptotic behavior of automorphism groups of function fields when genus tends to infinity. Motivated by applications in coding and cryptography, we consider the maximum size of abelian subgroups of the automorphism group $\mbox{Aut}(F/\mathbb{F}_q)$ in terms of genus ${g_F}$ for a function field $F$ over a finite field $\mathbb{F}_q$. Although the whole group $\mbox{Aut}(F/\mathbb{F}_q)$ could have size $Ω({g_F}^4)$, the maximum size $m_F$ of abelian subgroups of the automorphism group $\mbox{Aut}(F/\mathbb{F}_q)$ is upper bounded by $4g_F+4$ for $g_F\ge 2$. In the present paper, we study the asymptotic behavior of $m_F$ by defining $M_q=\limsup_{{g_F}\rightarrow\infty}\frac{m_F \cdot \log_q m_F}{g_F}$, where $F$ runs through all function fields over $\mathbb{F}_q$. We show that $M_q$ lies between $2$ and $3$ (or $4$) for odd characteristic (or for even characteristic, respectively). This means that $m_F$ grows much more slowly than genus does asymptotically. The second part of this paper is to study the maximum size $b_F$ of subgroups of $\mbox{Aut}(F/\mathbb{F}_q)$ whose order is coprime to $q$. The Hurwitz bound gives an upper bound $b_F\le 84(g_F-1)$ for every function field $F/\mathbb{F}_q$ of genus $g_F\ge 2$. We investigate the asymptotic behavior of $b_F$ by defining ${B_q}=\limsup_{{g_F}\rightarrow\infty}\frac{b_F}{g_F}$, where $F$ runs through all function fields over $\mathbb{F}_q$. Although the Hurwitz bound shows ${B_q}\le 84$, there are no lower bounds on $B_q$ in literature. One does not even know if ${B_q}=0$. For the first time, we show that ${B_q}\ge 2/3$ by explicitly constructing some towers of function fields in this paper.

math.NT↗

arXiv2017

Subspace Designs based on Algebraic Function Fields

Subspace designs are a (large) collection of high-dimensional subspaces $\{H_i\}$ of $\F_q^m$ such that for any low-dimensional subspace $W$, only a small number of subspaces from the collection have non-trivial intersection with $W$; more precisely, the sum of dimensions of $W \cap H_i$ is at most some parameter $L$. The notion was put forth by Guruswami and Xing (STOC'13) with applications to list decoding variants of Reed-Solomon and algebraic-geometric codes, and later also used for explicit rank-metric codes with optimal list decoding radius. Guruswami and Kopparty (FOCS'13, Combinatorica'16) gave an explicit construction of subspace designs with near-optimal parameters. This construction was based on polynomials and has close connections to folded Reed-Solomon codes, and required large field size (specifically $q \ge m$). Forbes and Guruswami (RANDOM'15) used this construction to give explicit constant degree "dimension expanders" over large fields, and noted that subspace designs are a powerful tool in linear-algebraic pseudorandomness. Here, we construct subspace designs over any field, at the expense of a modest worsening of the bound $L$ on total intersection dimension. Our approach is based on a (non-trivial) extension of the polynomial-based construction to algebraic function fields, and instantiating the approach with cyclotomic function fields. Plugging in our new subspace designs in the construction of Forbes and Guruswami yields dimension expanders over $\F^n$ for any field $\F$, with logarithmic degree and expansion guarantee for subspaces of dimension $Ω(n/(\log \log n))$.

cs.CC↗

arXiv2017

Efficiently list-decodable punctured Reed-Muller codes

The Reed-Muller (RM) code encoding $n$-variate degree-$d$ polynomials over ${\mathbb F}_q$ for $d < q$, with its evaluation on ${\mathbb F}_q^n$, has relative distance $1-d/q$ and can be list decoded from a $1-O(\sqrt{d/q})$ fraction of errors. In this work, for $d \ll q$, we give a length-efficient puncturing of such codes which (almost) retains the distance and list decodability properties of the Reed-Muller code, but has much better rate. Specificially, when $q =Ω( d^2/ε^2)$, we given an explicit rate $Ω\left(\fracε{d!}\right)$ puncturing of Reed-Muller codes which have relative distance at least $(1-ε)$ and efficient list decoding up to $(1-\sqrtε)$ error fraction. This almost matches the performance of random puncturings which work with the weaker field size requirement $q= Ω( d/ε^2)$. We can also improve the field size requirement to the optimal (up to constant factors) $q =Ω( d/ε)$, at the expense of a worse list decoding radius of $1-ε^{1/3}$ and rate $Ω\left(\frac{ε^2}{d!}\right)$. The first of the above trade-offs is obtained by substituting for the variables functions with carefully chosen pole orders from an algebraic function field; this leads to a puncturing for which the RM code is a subcode of a certain algebraic-geometric code (which is known to be efficiently list decodable). The second trade-off is obtained by concatenating this construction with a Reed-Solomon based multiplication friendly pair, and using the list recovery property of algebraic-geometric codes.

cs.IT↗

arXiv2017

Algebraic geometry codes with complementary duals exceed the asymptotic Gilbert-Varshamov bound

It was shown by Massey that linear complementary dual (LCD for short) codes are asymptotically good. In 2004, Sendrier proved that LCD codes meet the asymptotic Gilbert-Varshamov (GV for short) bound. Until now, the GV bound still remains to be the best asymptotical lower bound for LCD codes. In this paper, we show that an algebraic geometry code over a finite field of even characteristic is equivalent to an LCD code and consequently there exists a family of LCD codes that are equivalent to algebraic geometry codes and exceed the asymptotical GV bound.

cs.IT↗

arXiv2016

New MDS Self-Dual Codes from Generalized Reed-Solomon Codes

Both MDS and Euclidean self-dual codes have theoretical and practical importance and the study of MDS self-dual codes has attracted lots of attention in recent years. In particular, determining existence of $q$-ary MDS self-dual codes for various lengths has been investigated extensively. The problem is completely solved for the case where $q$ is even. The current paper focuses on the case where $q$ is odd. We construct a few classes of new MDS self-dual code through generalized Reed-Solomon codes. More precisely, we show that for any given even length $n$ we have a $q$-ary MDS code as long as $q\equiv1\bmod{4}$ and $q$ is sufficiently large (say $q\ge 2^n\times n^2)$. Furthermore, we prove that there exists a $q$-ary MDS self-dual code of length $n$ if $q=r^2$ and $n$ satisfies one of the three conditions: (i) $n\le r$ and $n$ is even; (ii) $q$ is odd and $n-1$ is an odd divisor of $q-1$; (iii) $r\equiv3\mod{4}$ and $n=2tr$ for any $t\le (r-1)/2$.

cs.IT↗

arXiv2016

On the List-Decodability of Random Self-Orthogonal Codes

In 2011, Guruswami-Håstad-Kopparty \cite{Gru} showed that the list-decodability of random linear codes is as good as that of general random codes. In the present paper, we further strengthen the result by showing that the list-decodability of random {\it Euclidean self-orthogonal} codes is as good as that of general random codes as well, i.e., achieves the classical Gilbert-Varshamov bound. Specifically, we show that, for any fixed finite field $\F_q$, error fraction $δ\in (0,1-1/q)$ satisfying $1-H_q(δ)\le \frac12$ and small $ε>0$, with high probability a random Euclidean self-orthogonal code over $\F_q$ of rate $1-H_q(δ)-ε$ is $(δ, O(1/ε))$-list-decodable. This generalizes the result of linear codes to Euclidean self-orthogonal codes. In addition, we extend the result to list decoding {\it symplectic dual-containing} codes by showing that the list-decodability of random symplectic dual-containing codes achieves the quantum Gilbert-Varshamov bound as well. This implies that list-decodability of quantum stabilizer codes can achieve the quantum Gilbert-Varshamov bound. The counting argument on self-orthogonal codes is an important ingredient to prove our result.

cs.IT↗

arXiv2015

Multipartite entangled states, symmetric matrices and error-correcting codes

A pure quantum state is called $k$-uniform if all its reductions to $k$-qudit are maximally mixed. We investigate the general constructions of $k$-uniform pure quantum states of $n$ subsystems with $d$ levels. We provide one construction via symmetric matrices and the second one through classical error-correcting codes. There are three main results arising from our constructions. Firstly, we show that for any given even $n\ge 2$, there always exists an $n/2$-uniform $n$-qudit quantum state of level $p$ for sufficiently large prime $p$. Secondly, both constructions show that their exist $k$-uniform $n$-qudit pure quantum states such that $k$ is proportional to $n$, i.e., $k=Ω(n)$ although the construction from symmetric matrices outperforms the one by error-correcting codes. Thirdly, our symmetric matrix construction provides a positive answer to the open question in \cite{DA} on whether there exists $3$-uniform $n$-qudit pure quantum state for all $n\ge 8$. In fact, we can further prove that, for every $k$, there exists a constant $M_k$ such that there exists a $k$-uniform $n$-qudit quantum state for all $n\ge M_k$. In addition, by using concatenation of algebraic geometry codes, we give an explicit construction of $k$-uniform quantum state when $k$ tends to infinity.

cs.IT↗

arXiv2015

A new class of rank-metric codes and their list decoding beyond the unique decoding radius

Compared with classical block codes, efficient list decoding of rank-metric codes seems more difficult. Although the list decodability of random rank-metric codes and limits to list decodability have been completely determined, little work on efficient list decoding rank-metric codes has been done. The only known efficient list decoding of rank-metric codes $\mC$ gives decoding radius up to the Singleton bound $1-R-\Ge$ with positive rate $R$ when $ρ(\mC)$ is extremely small, i.e., $Θ(\Ge^2)$ , where $ρ(\mC)$ denotes the ratio of the number of rows over the number of columns of $\mC$ \cite[STOC2013]{Guru2013}. It is commonly believed that list decoding of rank-metric codes $\mC$ with not small constant ratio $ρ(\mC)$ is hard. The main purpose of the present paper is to explicitly construct a class of rank-metric codes $\mC$ with not small constant ratio $ρ(\mC)$ and efficiently list decode these codes with decoding radius beyond $(1-R)/2$. Our key idea is to employ two-variable polynomials $f(x,y)$, where $f$ is linearized in variable $x$ and the variable $y$ is used to "fold" the code. In other words, rows are used to correct rank errors and columns are used to "fold" the code to enlarge decoding radius. Apart from the above algebraic technique, we have to prune down the list. The algebraic idea enables us to pin down the messages into a structured subspace of dimension linear in the number $n$ of columns. This "periodic" structure allows us to pre-encoding the messages to prune down the list. More precisely, we use subspace design introduced in \cite[STOC2013]{Guru2013} to get a deterministic algorithm with a larger constant list size and employ hierarchical subspace-evasive sets introduced in \cite[STOC2012]{Guru2012} to obtain a randomized algorithm with a smaller constant list size.

cs.IT↗

arXiv2015

An Improvement on the Hasse-Weil Bound and applications to Character Sums, Cryptography and Coding

The Hasse-Weil bound is a deep result in mathematics and has found wide applications in mathematics, theoretical computer science, information theory etc. In general, the bound is tight and cannot be improved. However, for some special families of curves the bound could be improved substantially. In this paper, we focus on the Hasse-Weil bound for the curve defined by $y^p-y=f(x)$ over the finite field $\F_q$, where $p$ is the characteristic of $\F_q$. Recently, Kaufman and Lovett \cite[FOCS2011]{KL11} showed that the Hasse-Weil bound can be improved for this family of curves with $f(x)=g(x)+h(x)$, where $g(x)$ is a polynomial of degree $\ll \sqrt{q}$ and $h(x)$ is a sparse polynomial of arbitrary degree but bounded weight degree. The other recent improvement by Rojas-Leon and Wan \cite[Math. Ann. 2011]{RW11} shows that an extra $\sqrt{p}$ can be removed for this family of curves if $p$ is very large compared with polynomial degree of $f(x)$ and $\log_pq$. In this paper, we show that the Hasse-Weil bound for this special family of curves can be improved if $q=p^n$ with odd $n$ which is the same case where Serre \cite{Se85} improved the Hasse-Weil bound. However, our improvement is greater than Serre's one for this special family of curves. Furthermore, our improvement works for small $p$ as well compared with the requirement of large $p$ by Rojas-Leon and Wan. In addition, our improvement finds interesting applications to character sums, cryptography and coding theory. The key idea behind is that this curve has the Hasse-Witt invariant $0$ and we show that the Hasse-Weil bound can be improved for any curves with the Hasse-Witt invariant $0$. The main tool used in our proof involves Newton polygon and some results in algebraic geometry.

cs.DM↗

arXiv2014

On the representability of the bi-uniform matroid

Every bi-uniform matroid is representable over all sufficiently large fields. But it is not known exactly over which finite fields they are representable, and the existence of efficient methods to find a representation for every given bi-uniform matroid has not been proved. The interest of these problems is due to their implications to secret sharing. The existence of efficient methods to find representations for all bi-uniform matroids is proved here for the first time. The previously known efficient constructions apply only to a particular class of bi-uniform matroids, while the known general constructions were not proved to be efficient. In addition, our constructions provide in many cases representations over smaller finite fields.

math.CO↗

arXiv2014

Some highlights of Harald Niederreiter's work

In this paper we give a short biography of Harald Niederreiter and we spotlight some cornerstones from his wide-ranging work. We focus on his results on uniform distribution, algebraic curves, polynomials and quasi-Monte Carlo methods. In the flavor of Harald's work we also mention some applications including numerical integration, coding theory and cryptography.

math.HO↗

arXiv2014

Torsion Limits and Riemann-Roch Systems for Function Fields and Applications

The Ihara limit (or -constant) $A(q)$ has been a central problem of study in the asymptotic theory of global function fields (or equivalently, algebraic curves over finite fields). It addresses global function fields with many rational points and, so far, most applications of this theory do not require additional properties. Motivated by recent applications, we require global function fields with the additional property that their zero class divisor groups contain at most a small number of $d$-torsion points. We capture this by the torsion limit, a new asymptotic quantity for global function fields. It seems that it is even harder to determine values of this new quantity than the Ihara constant. Nevertheless, some non-trivial lower- and upper bounds are derived. Apart from this new asymptotic quantity and bounds on it, we also introduce Riemann-Roch systems of equations. It turns out that this type of equation system plays an important role in the study of several other problems in areas such as coding theory, arithmetic secret sharing and multiplication complexity of finite fields etc. Finally, we show how our new asymptotic quantity, our bounds on it and Riemann-Roch systems can be used to improve results in these areas.

math.AG↗

arXiv2014

Erasure List-Decodable Codes from Random and Algebraic Geometry Codes

Erasure list decoding was introduced to correct a larger number of erasures with output of a list of possible candidates. In the present paper, we consider both random linear codes and algebraic geometry codes for list decoding erasure errors. The contributions of this paper are two-fold. Firstly, we show that, for arbitrary $0 0$ ($R$ and $ε$ are independent), with high probability a random linear code is an erasure list decodable code with constant list size $2^{O(1/ε)}$ that can correct a fraction $1-R-ε$ of erasures, i.e., a random linear code achieves the information-theoretic optimal trade-off between information rate and fraction of erasure errors. Secondly, we show that algebraic geometry codes are good erasure list-decodable codes. Precisely speaking, for any $0 0$, a $q$-ary algebraic geometry code of rate $R$ from the Garcia-Stichtenoth tower can correct $1-R-\frac{1}{\sqrt{q}-1}+\frac{1}{q}-ε$ fraction of erasure errors with list size $O(1/ε)$. This improves the Johnson bound applied to algebraic geometry codes. Furthermore, list decoding of these algebraic geometry codes can be implemented in polynomial time.

cs.IT↗

arXiv2013

Sequences with high nonlinear complexity

We improve lower bounds on the $k$th-order nonlinear complexity of pseudorandom sequences over finite fields and we establish a probabilistic result on the behavior of the $k$th-order nonlinear complexity of random sequences over finite fields.

cs.IT↗

arXiv2013

Euclidean and Hermitian Self-orthogonal Algebraic Geometry Codes and Their Application to Quantum Codes

In the present paper, we show that if the dimension of an arbitrary algebraic geometry code over a finite field of even characters is slightly less than half of its length, then it is equivalent to an Euclidean self-orthogonal code. However, in the literatures, a strong contrition about existence of certain differential is required to obtain such a result. We also show a similar result on Hermitian self-orthogonal algebraic geometry codes. As a consequence, we can apply our result to quantum codes and obtain quantum codes with good asymptotic bounds.

cs.IT↗

arXiv2013

A Construction of Quantum Codes via A Class of Classical Polynomial Codes

There have been various constructions of classical codes from polynomial valuations in literature \cite{ARC04, LNX01,LX04,XF04,XL00}. In this paper, we present a construction of classical codes based on polynomial construction again. One of the features of this construction is that not only the classical codes arisen from the construction have good parameters, but also quantum codes with reasonably good parameters can be produced from these classical codes. In particular, some new quantum codes are constructed (see Examples \ref{5.5} and \ref{5.6}).

cs.IT↗

arXiv2013

Quantum Gilbert-Varshamov Bound Through Symplectic Self-Orthogonal Codes

It is well known that quantum codes can be constructed through classical symplectic self-orthogonal codes. In this paper, we give a kind of Gilbert-Varshamov bound for symplectic self-orthogonal codes first and then obtain the Gilbert-Varshamov bound for quantum codes. The idea of obtaining the Gilbert-Varshamov bound for symplectic self-orthogonal codes follows from counting arguments.

cs.IT↗

arXiv2013

Optimal rate algebraic list decoding using narrow ray class fields

We use class field theory, specifically Drinfeld modules of rank 1, to construct a family of asymptotically good algebraic-geometric (AG) codes over fixed alphabets. Over a field of size $\ell^2$, these codes are within $2/(\sqrt{\ell}-1)$ of the Singleton bound. The functions fields underlying these codes are subfields with a cyclic Galois group of the narrow ray class field of certain function fields. The resulting codes are "folded" using a generator of the Galois group. This generalizes earlier work by the first author on folded AG codes based on cyclotomic function fields. Using the Chebotarev density theorem, we argue the abundance of inert places of large degree in our cyclic extension, and use this to devise a linear-algebraic algorithm to list decode these folded codes up to an error fraction approaching $1-R$ where $R$ is the rate. The list decoding can be performed in polynomial time given polynomial amount of pre-processed information about the function field. Our construction yields algebraic codes over constant-sized alphabets that can be list decoded up to the Singleton bound --- specifically, for any desired rate $R \in (0,1)$ and constant $\eps > 0$, we get codes over an alphabet size $(1/\eps)^{O(1/\eps^2)}$ that can be list decoded up to error fraction $1-R-\eps$ confining close-by messages to a subspace with $N^{O(1/\eps^2)}$ elements. Previous results for list decoding up to error-fraction $1-R-\eps$ over constant-sized alphabets were either based on concatenation or involved taking a carefully sampled subcode of algebraic-geometric codes. In contrast, our result shows that these folded algebraic-geometric codes {\em themselves} have the claimed list decoding property.

math.NT↗