arXiv subjects

Huawei Wu

Publications and source records attributed to Huawei Wu.

10 recordsLinked to original sources

arXiv2026

Norm-Query Complexity of Algorithmic Problems in Finite-Dimensional p-adic Normed Spaces

We study the deterministic norm-query complexity of computational problems in finite-dimensional vector spaces over $\mathbb{Q}_p$ equipped with an arbitrary ultrametric norm. For orthogonalization, we prove that no uniform finite query bound depending only on the dimension exists: for every deterministic algorithm that produces an $N$-orthogonal basis for every ultrametric norm $N$, the number of norm queries is unbounded as $N$ varies. We then study the Longest Vector Problem (LVP) for a rank-$m$ $p$-adic lattice. By adapting a brute-force search to the general norm-query setting and eliminating the scalar redundancy among nonzero coefficient vectors modulo $p$, we obtain an algorithm using exactly $(p^m-1)/(p-1)$ norm queries for $m\ge 2$, and prove that no deterministic norm-query algorithm can use fewer queries in the worst case. Finally, we consider the Closest Vector Problem (CVP). Apart from the trivial cases in which no norm query is needed, we prove that the deterministic worst-case norm-query complexity of the CVP is unbounded, even when the lattice and the target vector are fixed.

cs.CC↗

arXiv2026

Optimal Repair Bandwidth and Repair I/O of $(n,n-2,2)$ MDS Array Codes

We give a complete determination of the exact optimal worst-case repair bandwidth and repair I/O for linear exact repair of $(n,n-2,2)$ MDS array codes over every finite field $\mathbb{F}_q$ and for every admissible code length $3\le n\le q^2+1$. For repair bandwidth, we prove that the optimum is governed, up to a short explicit list of small exceptional cases, by the maximum of the sharpened $n$-only lower bound $\lceil(5n-8)/4\rceil$ and the projective counting, equivalently incidence-multiplicity, bound $2n-q-3$. For repair I/O, we obtain the analogous exact formula with $\lceil(4n-6)/3\rceil$ in place of $\lceil(5n-8)/4\rceil$, with the single special value at $n=4$. Thus, we completely resolve the first non-trivial redundancy and sub-packetization regime $(r,\ell)=(2,2)$ for both repair bandwidth and repair I/O.

cs.IT↗

arXiv2026

3-Designs from $\mathrm{GL}_2(\mathbb{F}_q)$-Invariant Subspaces of $\mathbb F_q[X,Y]_k$

We present a uniform framework for constructing $3$-designs from $\mathrm{GL}_2(\mathbb F_q)$-invariant subspaces of $\mathbb F_q[X,Y]_k$, the space of homogeneous polynomials of degree $k$. Given such a subspace $W$, we associate a $\mathrm{PGL}_2(\mathbb F_q)$-invariant family of $k$-subsets of $\mathbb P^1(\mathbb F_q)$. Whenever this family is nonempty, it forms a $3\text{-}(q+1,k,\lambda)$ design. Via the Cayley transform, the construction is reformulated on the unit circle $U_{q+1}\subseteq \mathbb F_{q^2}^{\times}$, where the block conditions become explicit linear relations among elementary symmetric polynomials. This reformulation unifies several previously disparate constructions and simplifies a number of delicate ad hoc computations. When $k\le q$, the evaluation map on $\mathbb P^1(\mathbb F_q)$ identifies $W$ with a subcode $C_W$ of the projective Reed--Solomon code. We show that the associated block family is nonempty if and only if $d(C_W)=q+1-k$. Under this condition, the supports of minimum-weight codewords in $C_W$, as well as the supports of suitable fixed-weight codewords in the dual code $C_W^\perp$, yield further $3$-designs. Applying this framework to the Lucas subspaces, which form a distinguished family of invariant subspaces, we obtain explicit block descriptions, classify the cases in which the defining conditions reduce to a single equation, and establish several emptiness and nonemptiness results. In particular, for $q=p^e$ and $k=p^m+1$, we show that the associated block family is nonempty if and only if $m\mid e$, in which case it yields the Steiner system $S(3,p^m+1,q+1)$. Finally, in the ternary case $p=3$ and $k=7$, we use the weight distribution of the ternary Melas code to determine the design parameters left undetermined by Xu et al.

math.CO↗

arXiv2026

The Incidence-Multiplicity Bound for Linear Exact Repair in MDS Array Codes

We study linear exact repair for $(n,k,\ell)$ MDS array codes over $\mathbb{F}_q$, with redundancy $r=n-k$, in the regime where $q$, $r$, and $\ell$ are fixed and the code length $n$ varies. A recent projective counting argument gives a general lower bound on repair bandwidth and repair I/O in this setting. While this bound is attained over a broad interval of code lengths in the two-parity case, it is not attained once $r\ge 3$ and $\ell\ge 2$. In this paper, we refine the counting argument behind this bound and establish a sharper lower bound, which we call the incidence-multiplicity bound. We prove that for every $(n,k,\ell)$ MDS array code over $\mathbb{F}_q$ with $r\ge 2$, both the average and worst-case repair bandwidth, as well as the average and worst-case repair I/O, are at least $$\ell(n-1)-(r-1)\frac{q^\ell-1}{q-1}.$$This bound agrees with the earlier projective counting bound when $r=2$, and is strictly stronger for every $r\ge 3$. We also show that the incidence-multiplicity bound is sharp in a broad parameter range. Assume that $\ell\ge 2$, $r\ge 2$, $(r-1)\mid(q-1)$, and $(q-1)/(r-1)\ge 2$. Then for every integer $n$ satisfying $$2(r-1)\frac{q^\ell-1}{q-1}\le n\le q^\ell+1,$$ there exists an $(n,n-r,\ell)$ MDS array code over $\mathbb{F}_q$ that attains the incidence-multiplicity bound simultaneously for both repair bandwidth and repair I/O. These codes arise from field reduction of a normal rational curve. Together, these results reveal incidence multiplicity as the governing geometric principle for linear exact repair in MDS array codes beyond the two-parity case.

cs.IT↗

arXiv2026

Linear Exact Repair in MDS Array Codes: A General Lower Bound and Its Attainability

For an $(n,k,\ell)$ MDS array code over $\mathbb{F}_q$, how small can the repair bandwidth and repair I/O be under linear exact repair? We study this question in the regime where the field size $q$, the redundancy $r=n-k$, and the sub-packetization level $\ell$ are fixed, while the code length $n$ varies, and we develop a geometric approach to this setting. Our starting point is an intrinsic reformulation of linear exact repair for MDS array codes in terms of subspace intersections and, for repair I/O, the projective point configurations induced by a parity-check realization. This viewpoint yields a simple projective counting argument establishing the general lower bound $$\beta_{\mathrm{avg}},\beta_{\max},\gamma_{\mathrm{avg}},\gamma_{\max}\;\ge\;\ell(n-1)-\frac{q^{(r-1)\ell}-1}{q-1}$$ for linear exact repair of every $(n,k,\ell)$ MDS array code over $\mathbb{F}_q$ with redundancy $r=n-k\ge 2$. To our knowledge, this is the first lower bound of this form that applies to arbitrary redundancy $r\ge 2$ and sub-packetization level $\ell$. At first glance, the projective counting bound appears rather coarse and therefore unlikely to be attained. We prove that this intuition is correct whenever $r\ge 3$ and $\ell\ge 2$. For $r=2$, the picture changes completely. Using Desarguesian spreads from finite geometry, we construct MDS array codes that attain the bound over a broad interval of code lengths, up to the maximum possible length $q^{\ell}+1$, and do so simultaneously for both repair bandwidth and repair I/O. In the smallest nontrivial case $(r,\ell)=(2,2)$, we also prove a converse within the regular-spread model. Together, these results identify a uniform obstruction governing linear exact repair and show that, in the two-parity case, this obstruction is tight.

cs.IT↗

arXiv2024

The Differential and Boomerang Properties of a Class of Binomials

Let $q$ be an odd prime power with $q\equiv 3\ ({\rm{mod}}\ 4)$. In this paper, we study the differential and boomerang properties of the function $F_{2,u}(x)=x^2\big(1+u\eta(x)\big)$ over $\mathbb{F}_{q}$, where $u\in\mathbb{F}_{q}^*$ and $\eta$ is the quadratic character of $\mathbb{F}_{q}$. We determine the differential uniformity of $F_{2,u}$ for any $u\in\mathbb{F}_{q}^*$ and determine the differential spectra and boomerang uniformity of the locally-APN functions $F_{2,\pm 1}$, thereby disproving a conjecture proposed in \cite{budaghyan2024arithmetization} which states that there exist infinitely many $q$ and $u$ such that $F_{2,u}$ is an APN function.

math.NT↗

arXiv2024

An in-depth study of the power function $x^{q+2}$ over the finite field $\mathbb{F}_{q^2}$: the differential, boomerang, and Walsh spectra, with an application to coding theory

Let $q = p^m$, where $p$ is an odd prime number and $m$ is a positive integer. In this paper, we examine the finite field $\mathbb{F}_{q^2}$, which consists of $q^2$ elements. We first present an alternative method to determine the differential spectrum of the power function $f(x) = x^{q+2}$ on $\mathbb{F}_{q^2}$, incorporating several key simplifications. This methodology provides a new proof of the results established by Man, Xia, Li, and Helleseth in Finite Fields and Their Applications 84 (2022), 102100, which not only completely determine the differential spectrum of $f$ but also facilitate the analysis of its boomerang uniformity. Specifically, we determine the boomerang uniformity of $f$ for the cases where $q \equiv 1$ or $3$ (mod $6$), with the exception of the scenario where $p = 5$ and $m$ is even. Furthermore, for $p = 3$, we investigate the value distribution of the Walsh spectrum of $f$, demonstrating that it takes on only four distinct values. Using this result, we derive the weight distribution of a ternary cyclic code with four Hamming weights. The article integrates refined mathematical techniques from algebraic number theory and the theory of finite fields, employing several ingredients, such as exponential sums, to explore the cryptographic analysis of functions over finite fields. They can be used to explore the differential/boomerang uniformity across a wider range of functions.

cs.CR↗

arXiv2023

DNA-Correcting Codes in DNA Storage Systems

In [1], the authors proposed a new model of DNA storage system that integrates all three steps of retrieval and introduced the concept of DNA-correcting codes, which guarantees that the output of the storage system can be decoded to the original data. They also gave necessary and sufficient conditions for DNA-correcting codes when the data part is free of errors. In this paper, we generalize their results to the general case.

cs.IT↗

arXiv2023

The Weight Distributions of Two Classes of Linear Codes From Perfect Nonlinear Functions

In this paper, we employ general results on the value distributions of perfect nonlinear functions from $\mathbb{F}_{p^m}$ to $\mathbb{F}_p$ together with a specific group action to give a unified approach to determining the weight distributions of two classes of linear codes over $\mathbb{F}_p$ constructed from perfect nonlinear functions, where $p$ is an odd prime number and $m\in\mathbb{N}_+$.

cs.IT↗

arXiv2023

Circular External Difference Families: Construction and Non-Existence

The circular external difference family and its strong version, which themselves are of independent combinatorial interest, were proposed as variants of the difference family to construct new unconditionally secure non-malleable threshold schemes. In this paper, we present new results regarding the construction and non-existence of (strong) circular external difference families, thereby solving several open problems on this topic.

math.CO↗