Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1812.02502

A Tight Rate Bound and Matching Construction for Locally Recoverable Codes with Sequential Recovery From Any Number of Multiple Erasures

S. B. Balaji·Ganesh R. Kini·P. Vijay Kumar
View original source ↗Read source document ↗Export metadata ↓

Abstract

By a locally recoverable code (LRC), we will in this paper, mean a linear code in which a given code symbol can be recovered by taking a linear combination of at most $r$ other code symbols with $r << k$. A natural extension is to the local recovery of a set of $t$ erased symbols. There have been several approaches proposed for the handling of multiple erasures. The approach considered here, is one of sequential recovery meaning that the $t$ erased symbols are recovered in succession, each time contacting at most $r$ other symbols for assistance in recovery. Under the constraint that each erased symbol be recoverable by contacting at most $r$ other code symbols, this approach is the most general and hence offers maximum possible code rate. We characterize the maximum possible rate of an LRC with sequential recovery for any $r \geq 3$ and $t$. We do this by first deriving an upper bound on code rate and then going on to construct a {\em binary} code that achieves this optimal rate. The upper bound derived here proves a conjecture made earlier relating to the structure (but not the exact form) of the rate bound. Our approach also permits us to deduce the structure of the parity-check matrix of a rate-optimal LRC with sequential recovery. The parity-check matrix in turn, leads to a graphical description of the code. The construction of a binary code having rate achieving the upper bound derived here makes use of this description. Interestingly, it turns out that a subclass of binary codes that are both rate and block-length optimal, correspond to graphs known as Moore graphs that are regular graphs having the smallest number of vertices for a given girth. A connection with Tornado codes is also made in the paper.

Explore related subjects

cs.ITmath.IT

Keep this discovery

BibTeXRIS

S. B. Balaji, Ganesh R. Kini, P. Vijay Kumar. 2018-12-06. A Tight Rate Bound and Matching Construction for Locally Recoverable Codes with Sequential Recovery From Any Number of Multiple Erasures. https://arxiv.org/abs/1812.02502

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Mathematical Theory of Pragmatic Information

We propose a pragmatic information theory unifying communication, control, and decision-making. Its core is the isoteleia mapping, formalizing equifinality: distinct semantic paths leading to the same optimal action are pragmatically equivalent. This induces a three-tier hierarchy of syntactic, semantic, and pragmatic information, each abstraction discarding task-irrelevant distinctions. We develop pragmatic entropy, up/down mutual information, channel capacity, and rate-distortion, and prove three coding theorems generalizing Shannon's classical results. We introduce pragmatic value (VoI) and cost (CoI) of information as decision-theoretic duals to rate-distortion and capacity, respectively, and formulate a Lagrangian dual framework for cross-layer optimization. The pragmatic efficiency bound $\mathcal{E}_p(\lambda)=\sup_R[\Phi_p(R)-\lambda\,\mathrm{CoI}_p(R)]$ quantifies the maximum net utility any resource-constrained intelligent system can extract, thereby establishing a fundamental behavioral capacity limit---generalizing Shannon's symbol-level capacity to goal-directed action. Extensions to continuous messages yield closed-form Gaussian expressions, while dynamic settings are addressed via a Bellman equation for sequential decision-making. This framework provides a rigorous foundation for task-oriented communication, networked control, autonomous systems, and embodied AI, shifting focus from symbol fidelity to the effectiveness of information in guiding actions, and offers a unified mathematical language for next-generation intelligent systems.

cs.IT↗
arXiv2026

Data Protection in Function-Correcting Symbol-Pair Codes: Redundancy Bounds and Protection Profiles

In several storage systems, including DNA storage and flash memory, errors affect neighbouring symbols jointly, and the Hamming metric does not adequately capture such error patterns. The symbol-pair read channel, introduced by Cassuto and Blaum~\cite{cassuto2011codes}, addresses this by reading consecutive pairs of symbols rather than individual symbols. Motivated by this, we introduce function-correcting symbol-pair codes with data protection (FCSPC-DP), which guarantee reliable recovery of a desired function of the message while simultaneously protecting the message itself against symbol-pair errors. We derive bounds on the optimal redundancy of such codes and establish a relationship with joint-pair distance matrices. We also give explicit constructions of FCSPC-DP for locally pair-bounded functions and symbol-pair weight functions. We introduce the pair-separation constant of a function, the minimum symbol-pair distance between messages sharing a function value, and show that when it is sufficiently large, data protection requires no additional redundancy: the optimal redundancy coincides with that of the corresponding code without data protection. Considering the symbol-pair analogue of the $\alpha$-distance graph, we introduce two code invariants, the generation profile and the disconnection threshold, and use them to characterise a code's protection properties. Relating the two metrics through these invariants yields upper and lower bounds on the symbol-pair threshold in terms of its Hamming counterpart, both of which are attained. We further extend the classical Plotkin and sphere-packing bounds to this setting.

cs.IT↗
arXiv2026

Physics of Information Geometry - Part II: Small-Step Active Inference on the Probability Simplex

This paper is the second in a two-part investigation of the physics of information geometry. While Part I develops a physical foundation for distributional motion on the probability simplex, the present paper studies how that framework manifests in active inference. The treatment is fully self-contained and does not require familiarity with Part I. We focus in particular on active inference through small distributional steps and the geometric structure induced by such local motion. Starting from an initial distribution, an agent evolves its belief state toward a final target distribution through a sequence of constrained updates. We define a relative free energy functional with respect to the preferred distribution and extend it to a relative potential energy analogous to the Helmholtz/Gibbs free-energy decomposition. The evolution is subject to a per-step kinetic constraint expressed through the Kullback-Leibler (KL) divergence between consecutive distributions, which serves as a discrete kinetic energy on the probability simplex. Using the information-geometric Pythagorean theorem on KL balls, we show that sufficiently small local moves dominate large direct jumps, and that greedy maximization of free-energy reduction is globally optimal under the kinetic constraint. This leads to a sequential variational principle in which the optimal trajectory minimizes the associated Lagrangian of the optimization problem. Similar to classical mechanics, the Lagrangian takes on the form as the difference between the kinetic and potential terms, establishing a least-action principle for distributional motion on the simplex. The resulting optimal update admits a closed form as an exponentially tilted version of the current distribution toward the preferred distribution, parametrized by an inverse-temperature-like multiplier. We further extend the framework to incorporate state-dependent geodesic...

cs.IT↗