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Xiumei Li

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math.NT Information Theory math.IT Computer Vision

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preprint2026arXiv

TAFA-GSGC: Group-wise Scalable Point Cloud Geometry Compression with Progressive Residual Refinement

Scalable compression is essential for bandwidth-adaptive transmission, yet most learned codecs are optimized for a fixed rate-distortion point, making rate adaptation costly due to re-encoding or maintaining multiple bitstreams. In this work, we propose TAFA-GSGC, a scalable learned point cloud geometry codec that enables multi-quality decoding from a single bitstream and a single trained model. TAFA-GSGC combines layered residual refinement with channel-group entropy coding, and introduces a Target-Aligned Feature Aggregation module to reduce cross-layer redundancy in enhancement residuals. Our framework supports up to 9 decodable quality levels with monotonic quality improvement as more subbitstreams are received, while maintaining strong compression efficiency. Compared with the PCGCv2 baseline, TAFA-GSGC demonstrates improved RD performance, achieving average BD-rate reductions of 4.99% and 5.92% in terms of D1-PSNR and D2-PSNR, respectively.

preprint2020arXiv

A proof of Sondow's conjecture on the Smarandache function

The Smarandache function of a positive integer $n$ , denoted by $S(n)$ , is defined to be the smallest positive integer $j$ such that $n$ divides the factorial $j!$ . In this note, we prove that for any fixed number $k > 1$ , the inequality $n^k < S(n)!$ holds for almost all positive integers $n$ . This confirms Sondow's conjecture which asserts that the inequality $n^2 < S(n)!$ holds for almost all positive integers $n$ .

preprint2020arXiv

Polynomial analogue of the Smarandache function

In the integer case, the Smarandache function of a positive integer $n$ is defined to be the smallest positive integer $k$ such that $n$ divides the factorial $k!$ . In this paper, we first define a natural order for polynomials in $\mathbb{F}_q[t]$ over a finite field $\mathbb{F}_q$ and then define the Smarandache function of a non-zero polynomial $f \in \mathbb{F}_q[t]$ , denoted by $S(f)$ , to be the smallest polynomial $g$ such that $f$ divides the Carlitz factorial of $g$ . In particular, we establish an analogue of a problem of Erd{\H o}s, which implies that for almost all polynomials $f$ , $S(f)=t^d$ , where $d$ is the maximal degree of the irreducible factors of $f$ .

preprint2020arXiv

Weight distributions of several families of 3-weight binary linear codes

The linear codes with a few weights have been applied widely in combinatorial designs, secret sharing, association schemes, authentication codes and strongly regular graphs. In this paper, we first correct an erroneous result about the exponential sum $\sum_{x\in \mathbb{F}_{2^{e}}}χ_1\left(ax^{2^α+1}+bx\right).$ Then, using the above exponential sum, we construct several families of binary linear codes of $3$ -weight and determine their weight distributions. Moreover, Most of them can be used in secret sharing schemes.

preprint2020arXiv

Weight hierarchies and weight distributions of a familiy of $p$ -ary linear codes

The weight distribution and weight hierarchy of linear codes are two important research topics in coding theory. In this paper, by choosing proper defining sets from inhomogeneous quadratic functions over $\mathbb{F}_{q}^{2},$ we construct a family of $3$ -weight $p$ -ary linear codes and determine their weight distributions and weight hierarchies. Most of the codes can be used in secret sharing schemes.

preprint2020arXiv

Weight hierarchies of a family of linear codes associated with degenerate quadratic forms

We restrict a degenerate quadratic form $f$ over a finite field of odd characteristic to subspaces. Thus, a quotient space related to $f$ is introduced. Then we get a non-degenerate quadratic form induced by $f$ over the quotient space. Some related results on the subspaces and quotient space are obtained. Based on this, we solve the weight hierarchies of a family of linear codes related to $f.$

preprint2013arXiv

Factoring integer using elliptic curves over rational number field $\mathbb{Q}$

For the integer $D=pq$ of the product of two distinct odd primes, we construct an elliptic curve $E_{2rD}:y^2=x^3-2rDx$ over $\mathbb Q$ , where $r$ is a parameter dependent on the classes of $p$ and $q$ modulo 8, and show, under the parity conjecture, that the elliptic curve has rank one and $v_p(x([k]Q))\not=v_q(x([k]Q))$ for odd $k$ and a generator $Q$ of the free part of $E_{2rD}(\mathbb Q)$ . Thus we can recover $p$ and $q$ from the data $D$ and $x([k]Q))$ . Furthermore, under the Generalized Riemann hypothesis, we prove that one can take $r<c\log^4D$ such that the elliptic curve $E_{2rD}$ has these properties, where $c$ is an absolute constant.

preprint2012arXiv

On the Selmer groups and Mordell-Weil groups of elliptic curves $y^{2} = x (x \pm p) (x \pm q)$ over imaginary quadratic number fields of class number one

Let $p$ and $q$ be odd prime numbers with $q - p = 2,$ the $φ-$ Selmer groups, Shafarevich-Tate groups ( $φ-$ and $2-$ part) and their dual ones as well the Mordell-Weil groups of elliptic curves $y^{2} = x (x \pm p) (x \pm q)$ over imaginary quadratic number fields of class number one are determined explicitly in many cases.

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Xiumei Li

Research overview

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8 publications

TAFA-GSGC: Group-wise Scalable Point Cloud Geometry Compression with Progressive Residual Refinement

A proof of Sondow's conjecture on the Smarandache function

Polynomial analogue of the Smarandache function

Weight distributions of several families of 3-weight binary linear codes

Weight hierarchies and weight distributions of a familiy of $p$ -ary linear codes

Weight hierarchies of a family of linear codes associated with degenerate quadratic forms

Factoring integer using elliptic curves over rational number field $\mathbb{Q}$

On the Selmer groups and Mordell-Weil groups of elliptic curves $y^{2} = x (x \pm p) (x \pm q)$ over imaginary quadratic number fields of class number one

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Xiumei Li

Research overview

Is this your work?

8 publications

TAFA-GSGC: Group-wise Scalable Point Cloud Geometry Compression with Progressive Residual Refinement

A proof of Sondow's conjecture on the Smarandache function

Polynomial analogue of the Smarandache function

Weight distributions of several families of 3-weight binary linear codes

Weight hierarchies and weight distributions of a familiy of ppp-ary linear codes

Weight hierarchies of a family of linear codes associated with degenerate quadratic forms

Factoring integer using elliptic curves over rational number field Q\mathbb{Q}Q

On the Selmer groups and Mordell-Weil groups of elliptic curves y2=x(x±p)(x±q) y^{2} = x (x \pm p) (x \pm q) y2=x(x±p)(x±q) over imaginary quadratic number fields of class number one

See the researcher in context

Building this map preview

Weight hierarchies and weight distributions of a familiy of $p$ -ary linear codes

Factoring integer using elliptic curves over rational number field $\mathbb{Q}$

On the Selmer groups and Mordell-Weil groups of elliptic curves $y^{2} = x (x \pm p) (x \pm q)$ over imaginary quadratic number fields of class number one