Kaimin Cheng

activity

20242026

collaborators

16 papers

math.NT2026

Counting subspaces with prescribed Welch sums

Kaimin Cheng, Jie Hu

Let $q = 2^{n}$ with $n = 2 m + 1$ for an integer $m \geq 1$ , and let $F_{q}$ be the finite field with $q$ elements. Let $W_{n} (x) = x^{2^{m} + 3}$ be the Welch function on $F_{q}$ . Fo…

math.NT2026

Counting zero-sum subspaces for the multiplicative inverse function

Kaimin Cheng

Let $q = 2^{n}$ with $n \geq 6$ , and let $F_{q}$ be the finite field with $q$ elements. For a subspace $E$ of $F_{q}$ , define \[ S(E)=\sum_{x\in E\setminus\{0\}}x^{-1}.\] Le…

math.NT2026

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

Kaimin Cheng

Let $s_{2} (n)$ be the binary sum-of-digits function and let $c_{t}$ be the natural density of the integers $n \geq 0$ for which $s_{2} (n + t) \geq s_{2} (n)$ . Earlier work of the author proved the…

math.CO2026

Deficit bounds and equality cases in the Tu--Deng problem

Kaimin Cheng

Let $N = 2^{k} - 1$ , and let $S_{t, k}$ consist of the pairs $0 \leq a, b < N$ such that $a + b \equiv t (mod N)$ and $\wt(a)+\wt(b)<k$. The Tu--Deng bound $∣ S_{t, k} ∣ \leq 2^{k - 1}$ has recently been…

math.NT2026

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

Kaimin Cheng

Let $q = 2^{e}$ with $e \geq 2$ even, put $d = (q^{2} + q + 1) /3$ , and let $σ (X) = X + X^{d} + X^{d q}$ be the permutation of $F_{q^{2}}$ introduced by Ding, Qu, Wang, Yuan, and Yuan. For $α\in\math…

math.NT2026

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

Kaimin Cheng

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<N:\ s_2…

math.NT2026

Banded quadratic digit functions along irreducible polynomials over finite fields

Kaimin Cheng

Let $q$ be an odd prime power and let $\F_q$ be the finite field with $q$ elements. Let $P (n)$ be the set of monic irreducible polynomials of degree $n$ over $\mathbb{F}_…

math.NT2026

On the $2$ -adic valuation of $σ_{k} (n)$

Kaimin Cheng, Ke Zhang

For a positive integer $k$ , let \[ σ_k(n)=\sum_{d\mid n} d^k \] be the divisor function of order $k$ , and let $ν_{p} (m)$ denote the $p$ -adic valuation of an integer $m$ . Motivated by…

math.NT2026

A proof of a permutation-inverse bent-function conjecture

Kaimin Cheng

Let $q = 2^{e}$ with $e$ even, and let $F_{q^{2}}$ be the finite field of order $q^{2}$ . Put $d = (q^{2} + q + 1) /3$ , and consider the permutation polynomial $$ σ(X)=X+X^d+X^{dq}\in\mathbb…

math.NT2025

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

Kaimin Cheng

Let $q$ be a prime power and $F_{q}$ the finite field with $q$ elements. For a positive integer $n$ , the binomial $X^{n} - 1 \in F_{q} [X]$ is said to be $3$ -sparse over…

math.NT2025

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

Kaimin Cheng

Let $q$ be a power of $2$ and let $F_{q}$ be the finite field with $q$ elements. For a positive integer $n$ , the polynomial $X^{n} - 1 \in F_{q} [X]$ is called $3$ -sparse ov…

cs.CR2025

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

Kaimin Cheng, Du Sheng

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 $e = ϕ (N)$ , and let $F_{q}$ be the finite field o…

math.NT2025

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

Kaimin Cheng, Arne Winterhof

Let $q$ be a prime power and $r$ a positive even integer. Let $F_{q}$ be the finite field with $q$ elements and $F_{q^{r}}$ be its extension field of degree $r$ . Le…

math.NT2024

The inverse stability of Artin-Schreier polynomials over finite fields

Kaimin Cheng

Let $p$ be a prime number and $q$ a power of $p$ . Let $F_{q}$ be the finite field with $q$ elements. For a positive integer $n$ and a polynomial $φ (X) \in F_{q} [X]$ , le…

cs.CR2024

A class of ternary codes with few weights

Kaimin Cheng

Let $ℓ^{m}$ be a power with $ℓ$ a prime greater than $3$ and $m$ a positive integer such that $3$ is a primitive root modulo $2 ℓ^{m}$ . Let $F_{3}$ be the finite field o…

math.NT2024

On binomial Weil sums and an application

Kaimin Cheng, Shuhong Gao

Let $p$ be a prime, and $N$ be a positive integer not divisible by $p$ . Denote by $ord_{N} (p)$ the multiplicative order of $p$ modulo $N$ . Let $F_{q}$ represent the fini…

Counting subspaces with prescribed Welch sums

Counting zero-sum subspaces for the multiplicative inverse function

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

Deficit bounds and equality cases in the Tu--Deng problem

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

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

Banded quadratic digit functions along irreducible polynomials over finite fields

On the 2-adic valuation of σk​(n)

A proof of a permutation-inverse bent-function conjecture

The 3-sparsity of Xn−1 over finite fields

The 3-sparsity of Xn−1 over finite fields, II

Weight distribution of a class of p-ary codes

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

The inverse stability of Artin-Schreier polynomials over finite fields

A class of ternary codes with few weights

On binomial Weil sums and an application

On the $2$ -adic valuation of $σ_{k} (n)$

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

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

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