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Sihuang Hu

Publications and source records attributed to Sihuang Hu.

At least 19 recordsLinked to original sources
arXiv2026

From Linear Exact Repair to Mixed-Alphabet Codes: The Bandwidth-Subpacketization Tradeoff of MDS Array Codes

MDS array codes achieve optimal storage efficiency in distributed storage systems, but their repair bandwidth depends critically on the sub-packetization level. While the theoretically optimal repair bandwidth (i.e., cut-set bound) of a high-rate exact-repair MDS array code is attainable with sufficiently large sub-packetization, the fundamental limits for fixed sub-packetization remain poorly understood. We study this problem for centralized linear exact repair of $h$ failures from $d$ helper nodes over a fixed finite field. We show that the rank profile of any linear repair scheme is constrained by two auxiliary mixed-alphabet codes. The first code is formed from the kernels of the helper projections and exists for all repair degrees; the second is formed from their row spaces and applies in the full-helper regime when $d\ge r$. These codes have complementary alphabet profiles, and their size and minimum-distance requirements are forced solely by the MDS property and repair correctness. This gives a unified route from classical coding-theoretic upper bounds to repair-bandwidth lower bounds. As a consequence, we obtain several explicit lower bounds via Singleton, Hamming, Plotkin, and Griesmer arguments, including known bounds as corollaries. We further formulate a Delsarte-type support-enumerator LP for mixed-alphabet codes and give an exact type-compressed implementation. For fixed sub-packetization \(\ell\), this yields a polynomial-time procedure for evaluating the resulting finite-parameter lower bound.

cs.IT↗
arXiv2026

The Norm Spectrum of Barnes-Wall Lattices Below Twice the Minimum

Let $Λ_m$ be the Barnes-Wall lattice of dimension $2^m$, normalized to have minimum squared norm $d_m:=2^{m-1}$. We determine its complete norm spectrum strictly between $d_m$ and $2d_m$, proving that, for every integer $m\ge1$, $\{\|x\|^2:x\inΛ_m,\ d_m<\|x\|^2<2d_m\}=\{2d_m-2^j:j\in\mathbb Z,\ \lceil m/2\rceil\le j\le m-2\}$. In particular, the norm spectrum in this interval is nonempty for $m\ge4$. The proof uses a Gaussian-integer model in which lattice automorphisms arising from Clifford unitaries transform a vector into a pair of coordinate subvectors in a lower-dimensional lattice. Divisibility of the differences of their squared norms and a matrix Parseval identity provide the arithmetic and averaging constraints needed for induction on $m$. Every listed norm is realized explicitly by a sum of two minimal vectors of the lattice.

math.NT↗
arXiv2026

Optimal Conversion Bandwidth for MDS Convertible Codes in the Split Regime with $r^F<k^F<r^I$

Erasure codes are widely used in distributed storage systems to provide fault tolerance. An $[n,k]$ erasure code encodes $k$ data symbols into $n$ coded symbols and distributes them across $n$ storage nodes. Once the code parameters are fixed, the achievable fault tolerance is also fixed. However, the failure rates of storage nodes may vary over time, and dynamically adapting the code parameters to these variations can substantially reduce storage overhead. Motivated by this observation, Maturana and Rashmi introduced convertible codes, which allow an $[n^I,k^I]$ initial code with redundancy $r^I=n^I-k^I$ to be transformed into an $[n^F,k^F]$ final code with redundancy $r^F=n^F-k^F$, while preserving the required code properties. Convertible codes whose initial and final codes are both MDS codes are called MDS convertible codes. They are of particular interest because MDS codes provide the maximum erasure tolerance for a given amount of storage overhead. Several works have established lower bounds and constructions for the bandwidth cost of MDS convertible codes in the split regime. However, the tight bound in the parameter range $r^F<k^F<r^I$ remains unknown. In this work, we establish a family of new rank inequalities for stable MDS convertible codes with linear conversion procedures and derive an improved lower bound on the bandwidth cost consisting of three cases for this remaining range. We prove that the bound is tight by presenting three explicit constructions, one for each case.

cs.IT↗
arXiv2026

Improved bounds for constant-power and low-power error-correcting cooling codes

Low-power error-correcting cooling (LPECC) codes and constant-power error-correcting cooling (CPECC) codes provide error correction while controlling power consumption and thermal effects in on-chip buses. In this paper, we study binary CPECC and LPECC codes with \(e=w-3\). For CPECC codes, we extend the applicability of the upper bound previously obtained by Zhao and Zhang from the quadratic-order condition \(w\ge 2t(t+1)+2\) to \(w\ge w_0(t)\), where \(w_0(t)\sim \sqrt{2}\,t^{3/2}\). Using Steiner systems, we show that the CPECC bound is attainable and asymptotically tight for fixed \(t,w\). For LPECC codes, we establish the new upper bound \(\left\lfloor\frac{\binom{n+2}{3}}{\binom{w+t}{3}}\right\rfloor\) for \(w\ge μ(t)\), where \(μ(t)\sim \sqrt{2}\,t^{3/2}\). This bound is strictly smaller than the previous bound of Zhao and Zhang whenever both apply, and is asymptotically tight for fixed \(t,w\) in the stated range.

cs.IT↗
arXiv2026

Divisibility of Trace Codes

A linear code is said to be $Δ$-divisible if the Hamming weights of all its codewords are divisible by $Δ$. The $p$-adic valuation of a code is defined as the greatest integer $t$ such that the code is $p^t$-divisible. In this paper, we establish a divisibility criterion for trace codes. Specifically, this criterion provides a systematic method to determine the $p$-adic valuation of the associated trace code, thereby extending Ward's classical divisibility criterion from standard generating sets (or matrices) to generalized generator matrices over an extension field. Furthermore, we present two applications of our framework. The first application provides a concise proof of the celebrated divisibility results on semisimple Abelian codes established by Delsarte and McEliece. The second application establishes several explicit lower bounds on the $p$-adic valuation of the number of solutions over $\mathbb{F}_{q^m}$ (where $q = p^e$) to the Artin--Schreier type equation $ f(x_1,\ldots,x_k)=y^q-y $. In particular, under the coprime condition $\left(d,\frac{q^m-1}{q-1}\right)=1$, we determine the exact minimum $p$-adic valuation of the number of solutions when $f$ is restricted to homogeneous polynomials of degree $d$.

math.CO↗
arXiv2026

Duality and Reverse Self-Dual Constructions for Hyperderivative Reed-Solomon Codes

Hyperderivative Reed-Solomon (HRS) codes form a class of maximum-distance-separable codes under the Niederreiter-Rosenbloom-Tsfasman metric and may be viewed as a derivative-evaluation extension of classical Reed-Solomon codes. For generalized Reed-Solomon codes, the Euclidean dual is again a generalized Reed-Solomon code. In this paper, we investigate the corresponding duality problem for HRS codes. Using a residue-theoretic argument, we derive an explicit component-wise representation for the Euclidean dual of an HRS code. The formula shows that, in general, the Euclidean dual is not an HRS code. Instead, it is blockwise upper-triangularly equivalent to a reverse-order HRS evaluation code, where the reversal occurs in the hyperderivative orders within each evaluation block. In particular, for full-domain HRS codes with low multiplicity, the triangular transformations reduce to diagonal scalings, and the Euclidean dual is obtained as the row reversal of an HRS code. Based on this reverse-order dual structure, we further study reverse self-dual HRS codes. We establish explicit criteria for reverse self-duality and construct several families from additive and multiplicative coset structures.

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 $3\text{-}(q+1,k,λ)$ 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

Lower Bounds on Conversion Bandwidth for MDS Convertible Codes in Split Regime

We propose several new lower bounds on the bandwidth costs of MDS convertible codes using a linear-algebraic framework. The derived bounds improve previous results in certain parameter regimes and match the bandwidth cost of the construction proposed by Maturana and Rashmi (2022 IEEE International Symposium on Information Theory) for $r^F\le r^I\le k^F$, implying that our bounds are tight in this case.

cs.IT↗
arXiv2026

Optimal Repair of $(k+2, k, 2)$ MDS Array Codes

Maximum distance separable (MDS) codes are widely used in distributed storage systems as they provide optimal fault tolerance for a given amount of storage overhead. The seminal work of Dimakis~\emph{et al.} first established a lower bound on the repair bandwidth for a single failed node of MDS codes, known as the \emph{cut-set bound}. MDS codes that achieve this bound are called minimum storage regenerating (MSR) codes. Numerous constructions and theoretical analyses of MSR codes reveal that they typically require exponentially large sub-packetization levels, leading to significant disk I/O overhead. To mitigate this issue, many studies explore the trade-offs between the sub-packetization level and repair bandwidth, achieving reduced sub-packetization at the cost of suboptimal repair bandwidth. Despite these advances, the fundamental question of determining the minimum repair bandwidth for a single failure of MDS codes with fixed sub-packetization remains open. In this paper, we address this challenge for the case of two parity nodes ($n-k=2$) and sub-packetization $\ell=2$. Under these parameters, we establish a correspondence between repair schemes and point sets on the projective line $\mathbb{P}^1$, and then derive a lower bound on repair bandwidth utilizing the sharply 3-transitive action of $\text{PGL}_2(\Fq)$. Furthermore, we extend this lower bound to the repair I/O, and construct two classes of explicit MDS array codes that achieve these bounds, offering practical code designs with provable repair efficiency.

cs.IT↗
arXiv2025

Making Wide Stripes Practical: Cascaded Parity LRCs for Efficient Repair and High Reliability

Erasure coding with wide stripes is increasingly adopted to reduce storage overhead in large-scale storage systems. However, existing Locally Repairable Codes (LRCs) exhibit structural limitations in this setting: inflated local groups increase single-node repair cost, multi-node failures frequently trigger expensive global repair, and reliability degrades sharply. We identify a key root cause: local and global parity blocks are designed independently, preventing them from cooperating during repair. We present Cascaded Parity LRCs (CP-LRCs), a new family of wide stripe LRCs that embed structured dependency between parity blocks by decomposing a global parity block across all local parity blocks. This creates a cascaded parity group that preserves MDS-level fault tolerance while enabling low-bandwidth single-node and multi-node repairs. We provide a general coefficient-generation framework, develop repair algorithms exploiting cascading, and instantiate the design with CP-Azure and CP-Uniform. Evaluations on Alibaba Cloud show reductions in repair time of up to 41% for single-node failures and 26% for two-node failures.

cs.DC↗
arXiv2025

Constructing $(h,d)$ cooperative MSR codes with sub-packetization $(d-k+h)(d-k+1)^{\lceil n/2 \rceil}$

We address the multi-node failure repair challenges for MDS array codes. Presently, two primary models are employed for multi-node repairs: the centralized model where all failed nodes are restored in a singular data center, and the cooperative model where failed nodes acquire data from auxiliary nodes and collaborate amongst themselves for the repair process.This paper focuses on the cooperative model, and we provide explicit constructions of optimal MDS array codes with $d$ helper nodes under this model. The sub-packetization level of our new codes is $(d-k+h)(d-k+1)^{\lceil n/2 \rceil}$ where $h$ is the number of failed nodes, $k$ the number of information nodes and $n$ the code length. This improves upon recent constructions given by Liu \emph{et al.} (IEEE Transactions on Information Theory, Vol. 69, 2023).

cs.IT↗
arXiv2025

Lower Bounds on the Sub-Packetization of Optimal-Access MSR Codes for Multiple-Node Repair

We establish lower bounds on the sub-packetization of optimal-access MSR codes in the context of multiple-node failures. These bounds generalize the tight bounds for single-node failure presented by Balaji et al. (IEEE Transactions on Information Theory, vol. 68, no. 10, 2022). Moreover, we utilize generating functions to provide a more refined analysis, further strengthening these bounds.

cs.IT↗
arXiv2024

Optimal $(2,δ)$ Locally Repairable Codes via Punctured Simplex Codes

Locally repairable codes (LRCs) have attracted a lot of attention due to their applications in distributed storage systems. In this paper, we provide new constructions of optimal $(2, δ)$-LRCs over $\mathbb{F}_q$ with flexible parameters. Firstly, employing techniques from finite geometry, we introduce a simple yet useful condition to ensure that a punctured simplex code becomes a $(2, δ)$-LRC. It is worth noting that this condition only imposes a requirement on the size of the puncturing set. Secondly, utilizing character sums over finite fields and Krawtchouk polynomials, we determine the parameters of more punctured simplex codes with puncturing sets of new structures. Several infinite families of LRCs with new parameters are derived. All of our new LRCs are optimal with respect to the generalized Cadambe-Mazumdar bound and some of them are also Griesmer codes or distance-optimal codes.

cs.IT↗
arXiv2023

MSR Codes with Linear Field Size and Smallest Sub-packetization for Any Number of Helper Nodes

The sub-packetization $\ell$ and the field size $q$ are of paramount importance in the MSR array code constructions. For optimal-access MSR codes, Balaji et al. proved that $\ell\geq s^{\left\lceil n/s \right\rceil}$, where $s = d-k+1$. Rawat et al. showed that this lower bound is attainable for all admissible values of $d$ when the field size is exponential in $n$. After that, tremendous efforts have been devoted to reducing the field size. However, till now, reduction to linear field size is only available for $d\in\{k+1,k+2,k+3\}$ and $d=n-1$. In this paper, we construct the first class of explicit optimal-access MSR codes with the smallest sub-packetization $\ell = s^{\left\lceil n/s \right\rceil}$ for all $d$ between $k+1$ and $n-1$, resolving an open problem in the survey (Ramkumar et al., Foundations and Trends in Communications and Information Theory: Vol. 19: No. 4). We further propose another class of explicit MSR code constructions (not optimal-access) with even smaller sub-packetization $s^{\left\lceil n/(s+1)\right\rceil }$ for all admissible values of $d$, making significant progress on another open problem in the survey. Previously, MSR codes with $\ell=s^{\left\lceil n/(s+1)\right\rceil }$ and $q=O(n)$ were only known for $d=k+1$ and $d=n-1$. The key insight that enables a linear field size in our construction is to reduce $\binom{n}{r}$ global constraints of non-vanishing determinants to $O_s(n)$ local ones, which is achieved by carefully designing the parity check matrices.

cs.IT↗
arXiv2023

ABS+ Polar Codes: Exploiting More Linear Transforms on Adjacent Bits

ABS polar codes were recently proposed to speed up polarization by swapping certain pairs of adjacent bits after each layer of polar transform. In this paper, we observe that applying the Arikan transform $(U_i, U_{i+1}) \mapsto (U_{i}+U_{i+1}, U_{i+1})$ on certain pairs of adjacent bits after each polar transform layer leads to even faster polarization. In light of this, we propose ABS+ polar codes which incorporate the Arikan transform in addition to the swapping transform in ABS polar codes. In order to efficiently construct and decode ABS+ polar codes, we derive a new recursive relation between the joint distributions of adjacent bits through different layers of polar transforms. Simulation results over a wide range of parameters show that the CRC-aided SCL decoder of ABS+ polar codes improves upon that of ABS polar codes by 0.1dB--0.25dB while maintaining the same decoding time. Moreover, ABS+ polar codes improve upon standard polar codes by 0.2dB--0.45dB when they both use the CRC-aided SCL decoder with list size $32$. The implementations of all the algorithms in this paper are available at https://github.com/PlumJelly/ABS-Polar

cs.IT↗
arXiv2023

Constructing MSR codes with subpacketization $2^{n/3}$ for $k+1$ helper nodes

Wang et al. (IEEE Transactions on Information Theory, vol. 62, no. 8, 2016) proposed an explicit construction of an $(n=k+2,k)$ Minimum Storage Regenerating (MSR) code with $2$ parity nodes and subpacketization $2^{k/3}$. The number of helper nodes for this code is $d=k+1=n-1$, and this code has the smallest subpacketization among all the existing explicit constructions of MSR codes with the same $n,k$ and $d$. In this paper, we present a new construction of MSR codes for a wider range of parameters. More precisely, we still fix $d=k+1$, but we allow the code length $n$ to be any integer satisfying $n\ge k+2$. The field size of our code is linear in $n$, and the subpacketization of our code is $2^{n/3}$. This value is slightly larger than the subpacketization of the construction by Wang et al. because their code construction only guarantees optimal repair for all the systematic nodes while our code construction guarantees optimal repair for all nodes.

cs.IT↗
arXiv2022

All the codeword symbols in polar codes have the same SER under the SC decoder

We consider polar codes constructed from the $2\times 2$ kernel $\begin{bmatrix} 1 & 0 \\ α& 1 \end{bmatrix}$ over a finite field $\mathbb{F}_{q}$, where $q=p^s$ is a power of a prime number $p$, and $α$ satisfies that $\mathbb{F}_{p}(α) = \mathbb{F}_{q}$. We prove that for any $\mathbb{F}_{q}$-symmetric memoryless channel, any code length, and any code dimension, all the codeword symbols in such polar codes have the same symbol error rate (SER) under the successive cancellation (SC) decoder.

cs.IT↗