arXiv subjects

Kaimin Cheng

Publications and source records attributed to Kaimin Cheng.

16 recordsLinked to original sources

arXiv2026

Sharp extremal asymptotics for Cusick's sum-of-digits bias at fixed Hamming weight

Let $s_2(n)$ be the binary sum-of-digits function and let $c_t$ be the natural density of the integers $n\ge0$ for which $s_2(n+t)\ge s_2(n)$. Earlier work of the author proved the universal exponential bound $$c_t-\frac12\ge 2^{-2s_2(t)-1},$$ thereby resolving Cusick's conjecture for every $t$. This estimate, however, does not reflect the true size of the smallest possible bias at a given large Hamming weight. In this paper, we determine this extremal scale sharply: $$\inf_{s_2(t)=k}\left(c_t-\frac12\right) \sim \frac{1}{2\sqrt\pi} \left(\frac{\log_2 k}{k}\right)^{3/2} \qquad(k\to\infty).$$ Thus the optimal fixed-weight gap is polynomial-logarithmic rather than exponential, with the explicit sharp leading constant $1/(2\sqrt\pi)$. The proof combines the five-cumulant Edgeworth expansion of Spiegelhofer and Wallner with a new extremal rigidity mechanism for near-extremal binary block patterns. We also prove a stability theorem for asymptotic extremizers and give a separate shadow-energy interpretation of the same constant.

math.NT↗

arXiv2026

Cyclic deletion rigidity and Macaulay shadows in the Tu--Deng problem

We determine all equality cases in the Tu--Deng bound $|S_{t,k}|\le 2^{k-1}$. If the $k$-bit cyclic word of $t$ has $R$ ones, $Z$ zeros, and cyclic one-gap lengths $g_1,\ldots,g_Z$, then equality holds if and only if $g_i\ge Z-1$ for every $i$. This resolves Conjecture~3.20 of Flori, Randriambololona, Cohen and Mesnager, and we also enumerate all equality parameters. For $R\ge Z$ we determine the sharp first stability gap and all extremal words, while for $R<Z$ we obtain an exact quantization of the deficit and an explicit run-sensitive lower bound. The proofs are structural: an explicit matrix conjugation identifies the auxiliary enumerators in the two recent complete proofs of the Tu--Deng conjecture. We then develop a rooted coarsening model for all coefficients, prove one-sided deletion rigidity and an exact Macaulay-flux identity, and derive a Macaulay--M\"obius formula from the bounded simplex at the highest cyclic level.

math.CO↗

arXiv2026

The complete cubic Walsh spectrum of a permutation-inverse Boolean family

Let $q=2^e$ with $e\ge2$ even, put $d=(q^2+q+1)/3$, and let $\sigma(X)=X+X^d+X^{dq}$ be the permutation of $\mathbb F_{q^2}$ introduced by Ding, Qu, Wang, Yuan, and Yuan. For $\alpha\in\mathbb F_q^*$, define the Boolean function \[ f_\alpha(x)=\operatorname{Tr}_{q^2}\bigl(\alpha(\sigma^{-1}(x))^3\bigr), \qquad x\in\mathbb F_{q^2}. \] In this paper, we determine the complete Walsh distribution of $f_\alpha$ in the remaining cubic case $\alpha\in(\mathbb F_q^*)^3$. More precisely, these functions are not bent but are $2$-plateaued: their Walsh values are precisely $0$ and $\pm 2q$, with exact multiplicities. The main new tool is a completion method for the outside Walsh coefficients: the punctured Fourier transform arising from the outside reduction is filled on the missing line, a modification invisible to outside frequencies, and the completed function is then identified with a Boolean component of a Kasami APN monomial. The APN property supplies a fourth-moment identity which, together with the known subfield spectrum and a Hasse divisibility congruence, forces the pointwise cubic spectrum.

math.NT↗

arXiv2026

A first-exit proof of Cusick's sum-of-digits conjecture

We prove Cusick's conjecture on the binary sum-of-digits function. More precisely, for every integer $t\ge 1$ we show that \[ c_t:=\lim_{N\to\infty}\frac{1}{N} \#\{0\le n \frac{1}{2}, \] and in fact obtain the explicit bound \[ c_t\ge \frac{1}{2}+2^{-2s_2(t)-1}, \] where $s_2(m)$ denotes the number of ones in the binary expansion of $m$. The proof is based on an exact deconvolution which replaces the distribution of $s_2(n+t)-s_2(n)$ by a finite stopped random-walk law. The required bias is then proved through first-exit medians for principal subsequence ideals.

math.NT↗

arXiv2026

Banded quadratic digit functions along irreducible polynomials over finite fields

Let $q$ be an odd prime power and let $\F_q$ be the finite field with $q$ elements. Let $\mathcal{P}(n)$ be the set of monic irreducible polynomials of degree $n$ over $\mathbb{F}_q$. For $f=t^n+f_{n-1}t^{n-1}+\cdots+f_0\in\mathcal{P}(n)$, fix coefficients $c_0,\ldots,c_m\in\mathbb{F}_q$ with $c_m\ne0$ and put $$ Q_A(f)=\sum_{j=0}^m c_j\sum_{i=j}^n f_i f_{i-j}+\ell_n(f),$$ where $\ell_n$ is an arbitrary linear form in the coefficients of $f$ and $f_n=1$. We prove that $Q_A$ is equidistributed on $\mathcal{P}(n)$: for every $\gamma\in\mathbb{F}_q$, $$\#\{f\in\mathcal{P}(n):Q_A(f)=\gamma\}=\frac{\#\mathcal{P}(n)}{q}+O_A(q^{19n/20+o(n)}),$$ as $n\to\infty$, with $q$ and the quadratic band fixed. This extends the finite-field Rudin--Shapiro result from nearest-neighbour correlations to arbitrary fixed symmetric Laurent symbols. The proof combines Vaughan's identity with rank estimates for Toeplitz forms; the main new ingredient is an averaged rank-defect estimate for reciprocal symbols in the central Type I range.

math.NT↗

arXiv2026

A proof of a permutation-inverse bent-function conjecture

Let $q=2^e$ with $e$ even, and let $\mathbb F_{q^2}$ be the finite field of order $q^2$. Put $d=(q^2+q+1)/3$, and consider the permutation polynomial $$ \sigma(X)=X+X^d+X^{dq}\in\mathbb F_{q^2}[X].$$ For $\alpha\in\mathbb F_q^*$, define $$f_{\alpha}(x)=\operatorname{Tr}_{q^2}\bigl(\alpha(\sigma^{-1}(x))^3\bigr),\qquad x\in\mathbb F_{q^2}.$$ We prove that $f_{\alpha}$ is bent if and only if $\alpha$ is not a cube in $\mathbb F_q$, thereby proving a conjecture of Li, Li, Helleseth, and Qu. The proof computes the Walsh values on $\mathbb F_q$ directly and treats the complementary parameters by reducing them to a two-variable exponential sum. A binary Hasse congruence, proved through finite carry analysis and a projective-frame cancellation of the only large carry component, forces the outside Walsh coefficients in the noncubic case to be $\pm q$. As an application, we identify a recent cyclotomic family of Xie, Li, Wang, and Zeng with the same construction in different coordinates and thereby prove their conjecture.

math.NT↗

arXiv2026

On the $2$-adic valuation of $\sigma_k(n)$

For a positive integer $k$, let \[ \sigma_k(n)=\sum_{d\mid n} d^k \] be the divisor function of order $k$, and let $\nu_p(m)$ denote the $p$-adic valuation of an integer $m$. Motivated by recent work on the $p$-adic valuation of $\sigma_k(n)$, we study $\nu_2(\sigma_k(n))$ in detail. We prove that, for every integer $n\ge 2$, \[ \nu_2(\sigma_k(n)) \le \begin{cases} \lceil \log_2 n \rceil, & \text{if $k$ is odd},\\[1mm] \lfloor \log_2 n \rfloor, & \text{if $k$ is even}. \end{cases} \] These bounds are best possible. More precisely, if $k$ is odd, then equality holds if and only if $n$ is a product of distinct Mersenne primes; if $k$ is even, then equality holds if and only if $n=3$. We also obtain an explicit formula for $\nu_2(\sigma_k(n))$ in terms of the prime factorization of $n$.

math.NT↗

arXiv2025

The $3$-sparsity of $X^n-1$ over finite fields, II

Let $q$ be a power of $2$ and let $\mathbb{F}_q$ be the finite field with $q$ elements. For a positive integer $n$, the polynomial $X^n-1\in\mathbb{F}_q[X]$ is called $3$-sparse over $\mathbb{F}_q$ if every monic irreducible factor of $X^n-1$ over $\mathbb{F}_q$ has at most three nonzero terms. This corrected version gives the characteristic-two classification. Writing $n=2^\lambda m$ with $m$ odd, $X^n-1$ is $3$-sparse over $\mathbb{F}_q$ if and only if either $\rad(m)\mid q^2-1$, or $q=2^e$, $3\nmid e$, and $m$ lies in the exceptional $7$-family \[ m=7^A s_0, \quad A\ge1, \quad (s_0,7)=1, \quad \rad(s_0)\mid q-1, \quad 3\nmid s_0/\gcd(s_0,q-1), \] with the additional maximal $7$-adic orbit condition $\ord_{7^a}(q)=3\cdot7^{a-1}$ for $1\le a\le A$. The latter condition is equivalent to $A=1$ or $7\nmid e$. This condition is necessary; for example, $X^{49}-1$ is not $3$-sparse over $\mathbb{F}_{128}$.

math.NT↗

arXiv2025

The $3$-sparsity of $X^n-1$ over finite fields

Let $q$ be a prime power and $\mathbb{F}_q$ the finite field with $q$ elements. For a positive integer $n$, the binomial $X^n - 1 \in \mathbb{F}_q[X]$ is said to be $3$-sparse over $\mathbb{F}_q$ if every irreducible factor of $X^n-1$ in $\mathbb{F}_q[X]$ is either a binomial or a trinomial. In 2021, Oliveira and Reis characterized all positive integers $n$ for which $X^n-1$ is $3$-sparse over $\mathbb{F}_q$ when $q = 2$ and $q = 4$, and raised the open problem of whether, for any given $q$, there are only finitely many primes $p$ such that $X^p-1$ is $3$-sparse over $\mathbb{F}_q$. In this paper, if $q$ is a power of an odd prime $r$, we then establish that for any positive integer not divisible by $r$, $X^n-1$ is $3$-sparse over $\mathbb{F}_q$ if and only if $n =p_1^{e_1} \cdots p_s^{e_s}$ for some nonnegative integers $e_1, \dots, e_s$, where $p_1, \dots, p_s$ are distinct prime divisors of $q^2 - 1$. This resolves the problem posed by Oliveira and Reis for odd characteristic.

math.NT↗

arXiv2025

Weight distribution of a class of $p$-ary codes

Let $p$ be a prime, and let $N$ be a positive integer such that $p$ is a primitive root modulo $N$. Define $q = p^e$, where $e = \phi(N)$, and let $\mathbb{F}_q$ be the finite field of order $q$ with $\mathbb{F}_p$ as its prime subfield. Denote by $\mathrm{Tr}$ the trace function from $\mathbb{F}_q$ to $\mathbb{F}_p$. For $\alpha \in \mathbb{F}_p$ and $\beta \in \mathbb{F}_q$, let $D$ be the set of nonzero solutions in $\mathbb{F}_q$ to the equation $\mathrm{Tr}(x^{\frac{q-1}{N}} + \beta x) = \alpha$. Writing $D = \{d_1, \ldots, d_n\}$, we define the code $\mathcal{C}_{\alpha,\beta} = \{(\mathrm{Tr}(d_1 x), \ldots, \mathrm{Tr}(d_n x)) : x \in \mathbb{F}_q\}$. In this paper, we investigate the weight distribution of $\mathcal{C}_{\alpha,\beta}$ for all $\alpha \in \mathbb{F}_p$ and $\beta \in \mathbb{F}_q$, with a focus on general odd primes $p$. When $\beta = 0$, we establish that $\mathcal{C}_{\alpha,0}$ is a two-weight code for any $\alpha \in \mathbb{F}_p$ and compute its weight distribution. For $\beta \neq 0$, we determine all possible weights of codewords in $\mathcal{C}_{\alpha,\beta}$, demonstrating that it has at most $p+1$ distinct nonzero weights. Additionally, we prove that the dual code $\mathcal{C}_{0,0}^{\perp}$ is optimal with respect to the sphere packing bound. These findings extend prior results to the broader case of any odd prime $p$.

cs.CR↗

arXiv2025

Multiplicative character sums over two classes of subsets of quadratic extensions of finite fields

Let $q$ be a prime power and $r$ a positive even integer. Let $\mathbb{F}_{q}$ be the finite field with $q$ elements and $\mathbb{F}_{q^r}$ be its extension field of degree $r$. Let $\chi$ be a nontrivial multiplicative character of $\mathbb{F}_{q^r}$ and $f(X)$ a polynomial over $\mathbb{F}_{q^r}$ with a simple root in $\mathbb{F}_{q^r}$. In this paper, we improve estimates for character sums $\sum\limits_{g \in\mathcal{G}}\chi(f(g))$, where $\mathcal{G}$ is either a subset of $\mathbb{F}_{q^r}$ of sparse elements, with respect to some fixed basis of $\mathbb{F}_{q^r}$ which contains a basis of $\mathbb{F}_{q^{r/2}}$, or a subset avoiding affine hyperplanes in general position. While such sums have been previously studied, our approach yields sharper bounds by reducing them to sums over the subfield $\mathbb{F}_{q^{r/2}}$ rather than sums over general linear spaces. These estimates can be used to prove the existence of primitive elements in $\mathcal{G}$ in the standard way.

math.NT↗

arXiv2024

The inverse stability of Artin-Schreier polynomials over finite fields

Let $p$ be a prime number and $q$ a power of $p$. Let $\mathbb{F}_q$ be the finite field with $q$ elements. For a positive integer $n$ and a polynomial $\varphi(X)\in\mathbb{F}_q[X]$, let $d_{n,\varphi}(X)$ denote the denominator of the $n$th iterate of $\frac{1}{\varphi(X)}$. The polynomial $\varphi(X)$ is said to be inversely stable over $\mathbb{F}_q$ if all polynomials $d_{n,\varphi}(X)$ are irreducible polynomial over $\mathbb{F}_q$ and distinct. In this paper, we characterize a class of inversely stable polynomials over $\mathbb{F}_q$. More precisely, for $\varphi(X)=X^{p^t}+aX+b\in\mathbb{F}_q[X]$ with $t$ being a positive integer, we provide a sufficient and necessary condition for $\varphi(X)$ to be inversely stable over $\mathbb{F}_q$.

math.NT↗

arXiv2024

A class of ternary codes with few weights

Let $\ell^m$ be a power with $\ell$ a prime greater than $3$ and $m$ a positive integer such that $3$ is a primitive root modulo $2\ell^m$. Let $\mathbb{F}_3$ be the finite field of order $3$, and let $\mathbb{F}$ be the $\ell^{m-1}(\ell-1)$-th extension field of $\mathbb{F}_3$. Denote by $\text{Tr}$ the absolute trace map from $\mathbb{F}$ to $\mathbb{F}_3$. For any $\alpha \in \mathbb{F}_3$ and $\beta \in\mathbb{F}$, let $D$ be the set of nonzero solutions in $\mathbb{F}$ to the equation $\text{Tr}(x^{\frac{q-1}{2\ell^m}} + \beta x) = \alpha$. In this paper, we investigate a ternary code $\mathcal{C}$ of length $n$, defined by $\mathcal{C} := \{(\text{Tr}(d_1x), \text{Tr}(d_2x), \dots, \text{Tr}(d_nx)) : x \in \mathbb{F}\}$ when we rewrite $D = \{d_1, d_2, \dots, d_n\}$. Using recent results on explicit evaluations of exponential sums, the Weil bound, and combinatorial techniques, we determine the Hamming weight distribution of the code $\mathcal{C}$. Furthermore, we show that when $\alpha = \beta =0$, the dual code of $\mathcal{C}$ is optimal with respect to the Hamming bound.

cs.CR↗

arXiv2024

On binomial Weil sums and an application

Let $p$ be a prime, and $N$ be a positive integer not divisible by $p$. Denote by ${\rm ord}_N(p)$ the multiplicative order of $p$ modulo $N$. Let $\mathbb{F}_q$ represent the finite field of order $q=p^{{\rm ord}_N(p)}$. For $a, b\in\mathbb{F}_q$, we define a binomial exponential sum by $$S_N(a,b):=\sum_{x\in\mathbb{F}_q\setminus\{0\}}\chi(ax^{\frac{q-1}{N}}+bx),$$ where $\chi$ is the canonical additive character of $\mathbb{F}_q$. In this paper, we provide an explicit evaluation of $S_{N}(a,b)$ for any odd prime $p$ and any $N$ satisfying ${\rm ord}_{N}(p)=\phi(N)$. Our elementary and direct approach allows for the construction of a class of ternary linear codes, with their exact weight distribution determined. Furthermore, we prove that the dual codes achieve optimality with respect to the sphere packing bound, thereby generalizing previous results from even to odd characteristic fields.

math.NT↗

arXiv2016

Permutational behavior of reversed Dickson polynomials over finite fields

In this paper, we use the method developed previously by Hong, Qin and Zhao to obtain several results on the permutational behavior of the reversed Dickson polynomial $D_{n,k}(1,x)$ of the $(k+1)$-th kind over the finite field ${\mathbb F}_{q}$. Particularly, we present the explicit evaluation of the first moment $\sum_{a\in {\mathbb F}_{q}}D_{n,k}(1,a)$. Our results extend the known results from the case $0\le k\le 3$ to the general $k\ge 0$ case.

math.NT↗

arXiv2016

Reversed Dickson polynomials of the fourth kind over finite fields

In this paper, we obtain several results on the permutational behavior of the reversed Dickson polynomial $D_{n,3}(1,x)$ of the fourth kind over the finite field ${\mathbb F}_{q}$. Particularly, we present the explicit evaluation of the first moment $\sum_{a\in {\mathbb F}_{q}}D_{n,3}(1,a)$.

math.NT↗