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Jiyou Li

Publications and source records attributed to Jiyou Li.

At least 19 recordsLinked to original sources
arXiv2026

Improving bounds for value sets of polynomials over finite fields

Let $\mathbb{F}_{q}$ be a finite field of characteristic $p$, and let $f \in \mathbb{F}_{q}[x]$ be a polynomial of degree $d > 0$. Denote the image set of this polynomial as $V_{f}=\{f(\alpha)\mid\alpha\in\mathbb{F}_{q}\}$ and denote the cardinality of this set as $N_{f}$. A much sharper bound for $N_{f}$ is established in this paper. In particular, for any $p\neq 2, 3$, and for nearly every generic quartic polynomial $f \in \mathbb{F}_{q}[x]$, we obtain $$\lvert N_f - \frac{5}{8} q \rvert \leq \frac{1}{2}\sqrt{q} + \frac{15}{4},$$ which holds as a simple corollary of the main result.

math.NT↗
arXiv2026

Probabilistic verification algorithm for linear codes

In this paper, we propose a probabilistic algorithm suitable for any linear code $C$ to determine whether a given vector $\mathbf{x}$ belongs to $ C$. The algorithm achieves $O(n\log n)$ time complexity, $ O(n^2)$ space complexity and with an error probability less than $1/\mathrm{poly}(n)$ in the asymptotic sense.

cs.IT↗
arXiv2025

On the biases and asymptotics of partitions with finite choices of parts

Biases in integer partitions have been studied recently. For three disjoint subsets $R,S,I$ of positive integers, let $p_{RSI}(n)$ be the number of partitions of $n$ with parts from $R\cup S\cup I$ and $p_{R>S,I}(n)$ be the number of such partitions with more parts from $R$ than that from $S$. In this paper, in the case that $R,S,I$ are finite we obtain a concrete formula of the asymptotic ratio of $p_{R>S,I}(n)$ to $p_{RSI}(n)$. We also propose a conjecture in the case that $R,S$ are certain infinite arithmetic progressions.

math.CO↗
arXiv2025

A Bijection between Necklaces and Restricted Multisets

We present a proof of Swee Hong Chan's conjecture establishing a bijection between the set of necklaces of length $n$ with at most $q$ colors, and the set of periodic functions $f: \mathbb{Z}_{n}\to {0, 1, ..., q-1}$ whose weighted sum is divisible by $n$, where $q$ and $n$ are coprime positive integers.

math.CO↗
arXiv2022

Improved error bounds for the distance distribution of Reed-Solomon codes

We use the generating function approach to derive simple expressions for the factorial moments of the distance distribution over Reed-Solomon codes. We obtain better upper bounds for the error term of a counting formula given by Li and Wan, which gives nontrivial estimates on the number of polynomials over finite fields with prescribed leading coefficients and a given number of linear factors. This improvement leads to new results on the classification of deep holes of Reed Solomon codes.

cs.IT↗
arXiv2021

A new sieve for restricted multiset counting

The Li--Wan sieve is extended to multisets when the underlying set is symmetric. The main ingredient of the proof is the Mobius inversion formula on the poset of partitions of $\{1,2,\dots,k\}$ ordered by refinement. As illustrative applications, we investigate the problems of partitions over finite fields and zero-sum multisets over the additive group $\mathbb{Z}/n\mathbb{Z}$. .

math.CO↗
arXiv2021

On sums of coefficients of Borwein type polynomials over arithmetic progressions

We obtain asymptotic formulas for sums over arithmetic progressions of coefficients of polynomials of the form $$\prod_{j=1}^n\prod_{k=1}^{p-1}(1-q^{pj-k})^s,$$ where $p$ is an odd prime and $n, s$ are positive integers. Let us denote by $a_i$ the coefficient of $q^i$ in the above polynomial and suppose that $b$ is an integer. We prove that $$\Big|\sum_{i\equiv b\ \text{mod}\ 2pn}a_i-\frac{v(b)p^{sn}}{2pn}\Big|\leq p^{sn/2},$$ where $v(b)=p-1$ if $b$ divisible by $p$ and $v(b)=-1$ otherwise. This improves a recent result of Goswami and Pantangi.

math.NT↗
arXiv2021

Newton Polygons of L-functions Associated to Deligne Polynomials

A conjecture of Le says that the Deligne polytope $Δ_d$ is generically ordinary if $p\equiv 1\ (\!\!\bmod\ D(Δ_d))$, where $D(Δ_d)$ is a combinatorial constant determined by $Δ_d$. In this paper a counterexample is given to show that the conjecture is not true in general.

math.NT↗
arXiv2020

Distinct coordinate solutions of linear equations over finite fields

Let $\mathbb{F}_q$ be the finite field of $q$ elements and $a_1,a_2, \ldots, a_k, b\in \mathbb{F}_q$. We investigate $N_{\mathbb{F}_q}(a_1, a_2, \ldots,a_k;b)$, the number of ordered solutions $(x_1, x_2, \ldots,x_k)\in\mathbb{F}_q^k$ of the linear equation $$ a_1x_1+a_2x_2+\cdots+a_kx_k=b$$ with all $x_i$ distinct. We obtain an explicit formula for $N_{\mathbb{F}_q}(a_1,a_2, \ldots, a_k;b)$ involving combinatorial numbers depending on $a_i$'s. In particular, we obtain closed formulas for two special cases. One is that $a_i, 1\leq i\leq k$ take at most three distinct values and the other is that $\sum_{i=1}^ka_i=0$ and $\sum_{i\in I}a_i\neq 0$ for any $I\subsetneq [k]$. The same technique works when $\mathbb{F}_q$ is replaced by $\mathbb{Z}_n$, the ring of integers modulo $n$. In particular, we give a new proof for the main result given by Bibak, Kapron and Srinivasan, which generalizes a theorem of Schönemann via a graph theoretic method.

math.NT↗
arXiv2019

A note on the Borwein conjecture

A conjecture of Borwein asserts that for any positive integers $n$ and $k$, the coefficient $a_{3k}$ of $q^{3k}$ in the expansion of $\prod_{j=0}^n (1-q^{3j+1})(1-q^{3j+2})$ is nonnegative. In this paper we prove that for any $0 \leq k\leq n$, there is a constant $0 0.$$

math.CO↗
arXiv2019

Distance Distribution to Received Words in Reed-Solomon Codes

Let $\mathbb{F}_q$ be the finite field of $q$ elements. In this paper we obtain bounds on the following counting problem: given a polynomial $f(x)\in \mathbb{F}_q[x]$ of degree $k+m$ and a non-negative integer $r$, count the number of polynomials $g(x)\in \mathbb{F}_q[x]$ of degree at most $k-1$ such that $f(x)+g(x)$ has exactly $r$ roots in $\mathbb{F}_q$. Previously, explicit formulas were known only for the cases $m=0, 1, 2$. As an application, we obtain an asymptotic formula on the list size of the standard Reed-Solomon code $[q, k, q-k+1]_q$.

math.NT↗
arXiv2017

On the construction of small subsets containing special elements in a finite field

In this note we construct a series of small subsets containing a non-d-th power element in a finite field by applying certain bounds on incomplete character sums. Precisely, let $h=\lfloor q^δ\rfloor>1$ and $d\mid q^h-1$. Let $r$ be a prime divisor of $q-1$ such that the largest prime power part of $q-1$ has the form $r^s$. Then there is a constant $0<ε<1$ such that for a ratio at least $ {q^{-εh}}$ of $α\in \mathbb{F}_{q^{h}} \backslash\mathbb{F}_{q}$, the set $S=\{ α-x^t, x\in\mathbb{F}_{q}\}$ of cardinality $1+\frac {q-1} {M(h)}$ contains a non-d-th power in $\mathbb{F}_{q^{\lfloor q^δ\rfloor}}$, where $t$ is the largest power of $r$ such that $t<\sqrt{q}/h$ and $M(h)$ is defined as $$M(h)=\max_{r \mid (q-1)} r^{\min\{v_r(q-1), \lfloor\log_r{q}/2-\log_r h\rfloor\}}.$$ Here $r$ runs thourgh prime divisors and $v_r(x)$ is the $r$-adic oder of $x$. For odd $q$, the choice of $δ=\frac 12-d, d=o(1)>0$ shows that there exists an explicit subset of cardinality $q^{1-d}=O(\log^{2+ε'}(q^h))$ containing a non-quadratic element in the field $\mathbb{F}_{q^h}$. On the other hand, the choice of $h=2$ shows that for any odd prime power $q$, there is an explicit subset of cardinality $1+\frac {q-1}{M(2)}$ containing a non-quadratic element in $\mathbb{F}_{q^2}$. This improves a $q-1$ construction by Coulter and Kosick \cite{CK} since $\lfloor \log_2{(q-1)}\rfloor\leq M(2) < \sqrt{q}$. In addition, we obtain a similar construction for small sets containing a primitive element. The construction works well provided $ϕ(q^h-1)$ is very small, where $ϕ$ is the Euler's totient function.

math.NT↗
arXiv2016

The Minimal and Maximal Sensitivity of the Simplified Weighted Sum Function

Sensitivity is an important complexity measure of Boolean functions. In this paper we present properties of the minimal and maximal sensitivity of the simplified weighted sum function. A simple close formula of the minimal sensitivity of the simplified weighted sum function is obtained. A phenomenon is exhibited that the minimal sensitivity of the weighted sum function is indeed an indicator of large primes, that is, for large prime number p, the minimal sensitivity of the weighted sum function is always equal to one.

cs.DM↗
arXiv2015

Counting polynomial subset sums

Let $D$ be a subset of a finite commutative ring $R$ with identity. Let $f(x)\in R[x]$ be a polynomial of positive degree $d$. For integer $0\leq k \leq |D|$, we study the number $N_f(D,k,b)$ of $k$-subsets $S\subseteq D$ such that \begin{align*} \sum_{x\in S} f(x)=b. \end{align*} In this paper, we establish several asymptotic formulas for $N_f(D,k, b)$, depending on the nature of the ring $R$ and $f$. For $R=\mathbb{Z}_n$, let $p=p(n)$ be the smallest prime divisor of $n$, $|D|=n-c \geq C_dn p^{-\frac 1d }+c$ and $f(x)=a_dx^d +\cdots +a_0\in \mathbb{Z}[x]$ with $(a_d, \dots, a_1, n)=1$. Then $$\left| N_f(D, k, b)-\frac{1}{n}{n-c \choose k}\right|\leq {δ(n)(n-c)+(1-δ(n))(C_dnp^{-\frac 1d}+c)+k-1\choose k},$$ partially answering an open question raised by Stanley \cite{St}, where $δ(n)=\sum_{i\mid n, μ(i)=-1}\frac 1 i$ and $C_d=e^{1.85d}$. Furthermore, if $n$ is a prime power, then $δ(n) =1/p$ and one can take $C_d=4.41$. For $R=\mathbb{F}_q$ of characteristic $p$, let $f(x)\in \mathbb{F}_q[x]$ be a polynomial of degree $d$ not divisible by $p$ and $D\subseteq \mathbb{F}_q$ with $|D|=q-c\geq (d-1)\sqrt{q}+c$. Then $$\left| N_f(D, k, b)-\frac{1}{q}{q-c \choose k}\right|\leq {\frac{q-c}{p}+\frac {p-1}{p}((d-1)q^{\frac 12}+c)+k-1 \choose k}.$$ If $f(x)=ax+b$, then this problem is precisely the well-known subset sum problem over a finite abelian group. Let $G$ be a finite abelian group and let $D\subseteq G$ with $|D|=|G|-c\geq c$. Then $$\left| N_x(D, k, b)-\frac{1}{|G|}{|G|-c \choose k}\right|\leq {c + (|G|-2c)δ(e(G))+k-1 \choose k},$$ where $e(G)$ is the exponent of $G$ and $δ(n)=\sum_{i\mid n, μ(i)=-1}\frac 1 i$. In particular, we give a new short proof for the explicit counting formula for the case $D=G$.

math.NT↗
arXiv2015

On the minimum distance of elliptic curve codes

Computing the minimum distance of a linear code is one of the fundamental problems in algorithmic coding theory. Vardy [14] showed that it is an \np-hard problem for general linear codes. In practice, one often uses codes with additional mathematical structure, such as AG codes. For AG codes of genus $0$ (generalized Reed-Solomon codes), the minimum distance has a simple explicit formula. An interesting result of Cheng [3] says that the minimum distance problem is already \np-hard (under \rp-reduction) for general elliptic curve codes (ECAG codes, or AG codes of genus $1$). In this paper, we show that the minimum distance of ECAG codes also has a simple explicit formula if the evaluation set is suitably large (at least $2/3$ of the group order). Our method is purely combinatorial and based on a new sieving technique from the first two authors [8]. This method also proves a significantly stronger version of the MDS (maximum distance separable) conjecture for ECAG codes.

cs.IT↗
arXiv2014

The simplified weighted sum function and its average sensitivity

In this paper we simplify the definition of the weighted sum Boolean function which used to be inconvenient to compute and use. We show that the new function has essentially the same properties as the previous one. In particular, the bound on the average sensitivity of the weighted sum Boolean function remains unchanged after the simplification.

cs.CC↗
arXiv2014

Asymptotic estimate for the polynomial coefficients

The polynomial coefficient $\binom {n,q}{k}$ is defined to be the coefficient of $x^{k}$ in the expansion of $(1+x+x^2+... +x^{q-1})^n$. In this note we give an asymptotic estimate for $\binom {n,q}{cn}$ as $n$ tends to infinity, where $c$ is a positive integer. Based on experimental results, it was conjectured that for any $n$, $\binom {n,q}{cn}-\binom {n,q-1}{cn}$ is unimodal and its maximum value occurs $q=\lfloor\log_{1+\frac 1{c}}{n}\rfloor$ or $q=\lfloor\log_{1+\frac 1{c}}{n}\rfloor+1$. In particular, when $c=1$, its maximum value occurs for $q=\lfloor\log_2{n}\rfloor$ or $q=\lfloor\log_2{n}\rfloor+1$.

math.CO↗