arXiv subjects

Yun Fan

Publications and source records attributed to Yun Fan.

At least 19 recordsLinked to original sources

arXiv2026

GUPO: Gradient Uncertainty-aware Policy Optimization for Post-Training Large Language Models

Group Relative Policy Optimization (GRPO) has become a widely used approach for post-training Large Language Models (LLMs) for reasoning. In GRPO, the group gradients induced by different queries within the same mini-batch are directly averaged to form the policy update. However, these group gradients can point in conflicting directions. Our empirical analysis suggests that group-gradient conflicts tend to be associated with less effective policy updates, motivating the need for a reliable aggregated update direction under such conflicts. Standard GRPO aggregation treats the realized group gradients as deterministic contributions and does not account for differences in their reliability during aggregation. To address this issue, we propose Gradient Uncertainty-Aware Policy Optimization (GUPO), which models each group gradient as a random variable under a Bayesian formulation and estimates its probability distribution. GUPO then derives gradient uncertainty using a Dirichlet-based formulation and uses it to calibrate the contribution of each group gradient during aggregation. Extensive experiments on multiple benchmarks demonstrate the effectiveness of GUPO.

cs.LG↗

arXiv2024

Self-dual 2-quasi Negacyclic Codes over Finite Fields

In this paper, we investigate the existence and asymptotic property of self-dual $2$-quasi negacyclic codes of length $2n$ over a finite field of cardinality $q$. When $n$ is odd, we show that the $q$-ary self-dual $2$-quasi negacyclic codes exist if and only if $q\,{\not\equiv}-\!1~({\rm mod}~4)$. When $n$ is even, we prove that the $q$-ary self-dual $2$-quasi negacyclic codes always exist. By using the technique introduced in this paper, we prove that $q$-ary self-dual $2$-quasi negacyclic codes are asymptotically good.

cs.IT↗

arXiv2024

Galois Self-dual 2-quasi Constacyclic Codes over Finite Fields

Let $F$ be a field with cardinality $p^\ell$ and $0\neq \lambda\in F$, and $0\le h<\ell$. Extending Euclidean and Hermitian inner products, Fan and Zhang introduced Galois $p^h$-inner product (DCC, vol.84, pp.473-492). In this paper, we characterize the structure of $2$-quasi $\lambda$-constacyclic codes over $F$; and exhibit necessary and sufficient conditions for $2$-quasi $\lambda$-constacyclic codes being Galois self-dual. With the help of a technique developed in this paper, we prove that, when $\ell$ is even, the Hermitian self-dual $2$-quasi $\lambda$-constacyclic codes are asymptotically good if and only if $\lambda^{1+p^{\ell/2}}=1$. And, when $p^\ell\,{\not\equiv}\,3~({\rm mod}~4)$, the Euclidean self-dual $2$-quasi $\lambda$-constacyclic codes are asymptotically good if and only if $\lambda^{2}=1$.

cs.IT↗

arXiv2023

Consta-dihedral Codes over Finite Fields

It is proved in a reference (Fan, Lin, IEEE TIT, vol.67, pp.5016-5025) that the self-dual (LCD respectively) dihedral codes over a finite field~$F$ with ${|F|=q}$ are asymptotically good if $q$ is even (odd respectively). In this paper, we investigate the algebraic property and the asymptotic property of conta-dihedral codes over $F$, and show that: if $q$ is even or $4\,|\,(q-1)$, then the self-dual consta-dihedral codes are asymptotically good; otherwise, the LCD consta-dihedral codes are asymptotically good. And, with the help of a technique developed in this paper, some errors in the reference mentioned above are corrected.

cs.IT↗

arXiv2022

Sharp Uncertainty Principle for Transitive $G$-Sets over Arbitrary Fields and Finite Groups

For any finite group $G$, any transitive $G$-set $X$ and any field ${\Bbb F}$, we consider the vector space ${\Bbb F}^X$ of all functions from $X$ to ${\Bbb F}$, which is a $G$-space isomorphic to the permutation ${\Bbb F} G$-module ${\Bbb F} X$. When the group algebra ${\Bbb F} G$ is semisimple and split, we find a specific basis $\widehat X$ of ${\Bbb F}^X$ and, for $f\in{\Bbb F}^X$, construct the Fourier transform $\widehat f\in{\Bbb F}^{\widehat X}$. We define the rank support $\mbox{rk-supp}(\widehat f)$ and prove that $\mbox{rk-supp}(\widehat f)=\dim {\Bbb F} G f$, where ${\Bbb F} G f$ is the submodule of ${\Bbb F} X$ generated by the element $f=\sum_{x\in X}f(x)x$. Next, we extend and strengthen the sharpened uncertainty principle for finite abelian groups, established by Feng, Hollmann, and Xiang in 2019, to a broader framework and a sharp version. For $0\ne f\in{\Bbb F}^X$, we construct a block $X_{{\rm supp}(f)}$ of $X$ and a subset ${\mathscr S}'^{-\!1}$ of $G$ determined by the support ${\rm supp}(f)$ of $f$, and show that $\dim{\Bbb F} Gf-\dim{\Bbb F}{\mathscr S}'^{-\!1}\!f\ge 1$ and $$ |{\rm supp}(f)|\cdot \dim{\Bbb F} Gf \ge |X|+ (\!\dim{\Bbb F} Gf-\dim{\Bbb F}{\mathscr S}'^{-1}f) \cdot|{\rm supp}(f)| -|X_{{\rm supp}(f)}|, $$ where ${\Bbb F}{\mathscr S}'^{-1}f$ denotes the subspace of ${\Bbb F}X$ spanned by the subset ${\mathscr S}'^{-1}f=\{\alpha f\,|\,\alpha\in{\mathscr S}'^{-1}\}\subseteq{\Bbb F} X$. We provide necessary and sufficient conditions for the above inequality to achieve equality. As corollaries, we derive many sharpened or classical versions of the finite-dimensional uncertainty principle, address an open question posed by Feng, Hollmann, and Xiang. When $|G|$ is a prime and $X=G$, we give a lower bound on $\dim {\Bbb F}Gf$ that recovers Tao's 2005 strong uncertainty principle, along with a precise characterization of the equality case.

math.GR↗

arXiv2022

Double Constacyclic Codes over Two Finite Commutative Chain Rings

Many kinds of codes which possess two cycle structures over two special finite commutative chain rings, such as ${\Bbb Z}_2{\Bbb Z}_4$-additive cyclic codes and quasi-cyclic codes of fractional index etc., were proved asymptotically good. In this paper we extend the study in two directions: we consider any two finite commutative chain rings with a surjective homomorphism from one to the other, and consider double constacyclic structures. We construct an extensive kind of double constacyclic codes over two finite commutative chain rings. And, developing a probabilistic method suitable for quasi-cyclic codes over fields, we prove that the double constacyclic codes over two finite commutative chain rings are asymptotically good.

cs.IT↗

arXiv2022

Asymptotic Properties of Quasi-Group Codes

This is a manuscript of a chapter prepared for a book. The good codes possess large information length and large minimum distance. A class of codes is said to be asymptotically good if there exists a positive real $\delta$ such that, for any positive integer $N$ we can find a code in the class with code length greater than $N$, and with both the rate and the relative minimum distance greater than $\delta$. The linear codes over any finite field are asymptotically good. More interestingly, the (asymptotic) GV-bound is a phase transition point for the linear codes; i.e., asymptotically speaking, the parameters of most linear codes attain the GV-bound. It is a long-standing open question: whether or not the cyclic codes over a finite field (which are an important class of codes) are asymptotically good? However, from a long time ago the quasi-cyclic codes of index $2$ were proved to be asymptotically good. This chapter consists of some of our studies on the asymptotic properties of several classes of quasi-group codes. We'll explain the studies in a consistent and self-contained style. We begin with the classical results on linear codes. In many cases we consider the quasi-group codes over finite abelian groups (including the cyclic case as a subcase of course), and study their asymptotic properties along two directions: (1) the order of the group (the coindex) is fixed while the index is going to infinity; (2) the index is small while the order of the group (the coindex) is going to infinity. Finally we describe the story on dihedral codes. The dihedral groups are non-abelian but near to cyclic groups (they have cyclic subgroups of index $2$). The asymptotic goodness of binary dihedral codes was obtained in the beginning of this century, and extended to the general dihedral codes recently.

cs.IT↗

arXiv2022

Self-dual 2-quasi-cyclic Codes and Dihedral Codes

We characterize the structure of 2-quasi-cyclic codes over a finite field F by the so-called Goursat Lemma. With the characterization, we exhibit a necessary and sufficient condition for a 2-quasi-cyclic code being a dihedral code. And we obtain a necessary and sufficient condition for a self-dual 2-quasi-cyclic code being a dihedral code (if charF = 2), or a consta-dihedral code (if charF odd). As a consequence, any self-dual 2-quasi-cyclic code generated by one element must be (consta-)dihedral. In particular, any self-dual double circulant code must be (consta-)dihedral. Also, we show a necessary and sufficient condition that the three classes (the self-dual double circulant codes, the self-dual 2-quasi-cyclic codes, and the self-dual (consta-)dihedral codes) are coincide each other.

cs.IT↗

arXiv2021

Self-dual $2$-quasi-abelian Codes

A kind of self-dual quasi-abelian codes of index $2$ over any finite field $F$ is introduced. By counting the number of such codes and the number of the codes of this kind whose relative minimum weights are small, such codes are proved to be asymptotically good provided $-1$ is a square in $F$. Moreover, a kind of self-orthogonal quasi-abelian codes of index $2$ are defined; and such codes always exist. In a way similar to that for self-dual quasi-abelian codes of index $2$, it is proved that the kind of the self-orthogonal quasi-abelian codes of index $2$ is asymptotically good.

cs.IT↗

arXiv2021

Constacyclic Codes over Commutative Finite Principal Ideal Rings

For any constacyclic code over a finite commutative chain ring of length coprime to the characteristic of the ring, we construct explicitly generator polynomials and check polynomials, and exhibit a BCH bound for such constacyclic codes. As a consequence, such constacyclic codes are principal. Further, we get a necessary and sufficient condition that the cyclic codes over a finite commutative principal ideal ring are all principal. This condition is still sufficient for constacyclic codes over such rings being principal.

cs.IT↗

arXiv2020

Dihedral group codes over finite fields

Bazzi and Mitter [3] showed that binary dihedral group codes are asymptotically good. In this paper we prove that the dihedral group codes over any finite field with good mathematical properties are asymptotically good. If the characteristic of the field is even, we construct asymptotically good self-dual dihedral group codes. If the characteristic of the filed is odd, we construct both the asymptotically good self-orthogonal dihedral group codes, and the asymptotically good LCD dihedral group codes.

cs.IT↗

arXiv2019

$\mathbb{Z}_2\mathbb{Z}_4$-Additive Cyclic Codes Are Asymptotically Good

We construct a class of $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes generated by pairs of polynomials, study their algebraic structures, and obtain the generator matrix of any code in the class. Using a probabilistic method, we prove that, for any positive real number $\delta<1/3$ such that the entropy at $3\delta/2$ is less than $1/2$, the probability that the relative minimal distance of a random code in the class is greater than $\delta$ is almost $1$; and the probability that the rate of the random code equals to $1/3$ is also almost $1$. As an obvious consequence, the $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes are asymptotically good.

cs.IT↗

arXiv2017

Galois LCD Codes over Finite Fields

In this paper, we study the complementary dual codes in more general setting (which are called Galois LCD codes) by a uniform method. A necessary and sufficient condition for linear codes to be Galois LCD codes is determined, and constacyclic codes to be Galois LCD codes are characterized. Some illustrative examples which constacyclic codes are Galois LCD MDS codes are provided as well. In particular, we study Hermitian LCD constacyclic codes. Finally, we present a construction of a class of Hermitian LCD codes which are also MDS codes.

cs.IT↗

arXiv2016

Isometrically Self-dual Cyclic Codes

General isometries of cyclic codes, including multipliers and translations, are introduced; and isometrically self-dual cyclic codes are defined. In terms of Type-I duadic splittings given by multipliers and translations, a necessary and sufficient condition for the existence of isometrically self-dual cyclic codes is obtained. A program to construct isometrically self-dual cyclic codes is provided, and illustrated by several examples. In particular, a class of isometrically self-dual MDS cyclic codes, which are alternant codes from a class of generalized Reed-Solomon codes, is presented.

cs.IT↗

arXiv2016

Permutation-like Matrix Groups with a Maximal Cycle of Length Power of Two

If every element of a matrix group is similar to a permutation matrix, then it is called a permutation-like matrix group. References [4], [5] and [6] showed that, if a permutation-like matrix group contains a maximal cycle such that the maximal cycle generates a normal subgroup and the length of the maximal cycle equals to a prime, or a square of a prime, or a power of an odd prime, then the permutation-like matrix group is similar to a permutation matrix group. In this paper, we prove that if a permutation-like matrix group contains a maximal cycle such that the maximal cycle generates a normal subgroup and the length of the maximal cycle equals to any power of 2, then it is similar to a permutation matrix group.

math.GR↗

arXiv2016

Nonlinear functions and difference sets on group actions

Let $G$, $H$ be finite groups and let $X$ be a finite $G$-set. $G$-perfect nonlinear functions from $X$ to $H$ have been studied in several papers. They have more interesting properties than perfect nonlinear functions from $G$ itself to $H$. By introducing the concept of a $(G, H)$-related difference family of $X$, we obtain a characterization of $G$-perfect nonlinear functions on $X$. When $G$ is abelian, we characterize a $G$-difference set of $X$ by the Fourier transform on a normalized $G$-dual set $\widehat X$. We will also investigate the existence and constructions of $G$-perfect nonlinear functions and $G$-bent functions. Several known results in [2,6,10,17] are direct consequences of our results.

math.CO↗

arXiv2016

Double Circulant Matrices

Double circulant matrices are introduced and studied. A formula to compute the rank r of a double circulant matrix is exhibited; and it is shown that any consecutive r rows of the double circulant matrix are linearly independent. As a generalization, multiple circulant matrices are also introduced. Two questions on square double circulant matrices are suggested.

math.RA↗

arXiv2015

Galois Self-Dual Constacyclic Codes

Generalizing Euclidean inner product and Hermitian inner product, we introduce Galois inner products, and study the Galois self-dual constacyclic codes in a very general setting by a uniform method. The conditions for existence of Galois self-dual and isometrically Galois self-dual constacyclic codes are obtained. As consequences, the results on self-dual, iso-dual and Hermitian self-dual constacyclic codes are derived.

cs.IT↗