Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1608.01738

Linear Network Coding over Rings, Part I: Scalar Codes and Commutative Alphabets

Joseph Connelly·Kenneth Zeger
View original source ↗Read source document ↗Export metadata ↓

Abstract

Fixed-size commutative rings are quasi-ordered such that all scalar linearly solvable networks over any given ring are also scalar linearly solvable over any higher-ordered ring. As consequences, if a network has a scalar linear solution over some finite commutative ring, then (i) the network is also scalar linearly solvable over a maximal commutative ring of the same size, and (ii) the (unique) smallest size commutative ring over which the network has a scalar linear solution is a field. We prove that a commutative ring is maximal with respect to the quasi-order if and only if some network is scalar linearly solvable over the ring but not over any other commutative ring of the same size. Furthermore, we show that maximal commutative rings are direct products of certain fields specified by the integer partitions of the prime factor multiplicities of the maximal ring's size. Finally, we prove that there is a unique maximal commutative ring of size $m$ if and only if each prime factor of $m$ has multiplicity in $\{1,2,3,4,6\}$. In fact, whenever $p$ is prime and $k \in \{1,2,3,4,6\}$, the unique such maximal ring of size $p^k$ is the field $GF(p^k)$. However, for every field $GF(p^k)$ with $k\not\in \{1,2,3,4,6\}$, there is always some network that is not scalar linearly solvable over the field but is scalar linearly solvable over a commutative ring of the same size. These results imply that for scalar linear network coding over commutative rings, fields can always be used when the alphabet size is flexible, but alternative rings may be needed when the alphabet size is fixed.

Explore related subjects

cs.ITmath.IT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Joseph Connelly, Kenneth Zeger. 2016-08-08. Linear Network Coding over Rings, Part I: Scalar Codes and Commutative Alphabets. https://doi.org/10.1109/tit.2017.2697421

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

KEEP EXPLORING

Related papers

arXiv2026

Radiance-Field Guided Pretraining: Scaling Localization Models with Unlabeled Wireless Signals

Radio frequency (RF)-based indoor localization offers significant promise for applications such as indoor navigation, augmented reality, and pervasive computing. While deep learning has greatly enhanced localization accuracy and robustness, existing localization models still face major challenges in cross-scene generalization due to their reliance on scene-specific labeled data. To address this, we introduce Radiance-Field Reinforced Pretraining (RFRP). This novel self-supervised pretraining framework couples a large localization model (LM) with a neural radio-frequency radiance field (RF-NeRF) in an asymmetrical autoencoder architecture. In this design, the LM encodes received RF spectra into latent, position-relevant representations, while the RF-NeRF decodes them to reconstruct the original spectra. This alignment between input and output enables effective representation learning using large-scale, unlabeled RF data, which can be collected continuously with minimal effort. To this end, we collected RF samples at 7,327,321 positions across 100 diverse scenes using four common wireless technologies--RFID, BLE, WiFi, and IIoT. Data from 75 scenes were used for training, and the remaining 25 for evaluation. Experimental results show that the RFRP-pretrained LM reduces localization error by over 40% compared to non-pretrained models and by 21% compared to those pretrained using supervised learning.

cs.IT↗
arXiv2026

Uniform Recovery of Structured Signals from Nonlinear Observations: Improved Error Rates

Consider the recovery of structured signals from nonlinear observations. Under Gaussian matrix and a large class of unknown nonlinear link functions, Plan and Vershynin (2016) showed that generalized Lasso achieves accurate nonuniform recovery of a fixed signal. More recently, Genzel and Stollenwerk (2023) showed that generalized Lasso is indeed capable of accurately recovering all structured signals. However, in some canonical settings with discontinuous link functions, their uniform recovery error rate is essentially slower than the nonuniform one. Specifically, in the recovery of $n$-dimensional $k$-sparse vectors from $m$ measurements, generalized Lasso with a perfectly tuned $\ell_1$ constraint achieves nonuniform error rate $ O(\sqrt{k\log(en/k)/m})$, while the uniform error rate of Genzel and Stollenwerk is no faster than $O((k\log(en/k)/m)^{1/4})$. In this paper, we narrow this gap by establishing improved uniform recovery guarantees under piecewise Lipschitz link functions with well-separated jump discontinuities. We analyze a projected gradient descent (PGD) algorithm whose projection can be onto a convex set or a cone, and our results for the PGD with a convex set are also valid for the generalized Lasso. In sparse recovery, the improved uniform error rates match the nonuniform rate $O(\sqrt{k\log(en/k)/m})$ up to logarithmic factors. Under the sign link function, we further show that iterative hard thresholding (a specific instance of the PGD) achieves uniform recovery error rate $O(\sqrt{k\log(en/k)/m})$, matching the nonuniform rate up to a universal constant. Technically, the uniform guarantees for the PGD are obtained by showing that the gradient maps satisfy the restricted approximate invertibility condition uniformly over all signals. We demonstrate that this is a general approach to uniform recovery under nonlinear observations.

cs.IT↗
arXiv2026

Recursive overlap Bernoulli distributions and an entropy concavity conjecture

We introduce a family of recursively generated finite probability distributions obtained from left and right embeddings with overlaps. The construction interpolates between the classical binomial distribution and the non-overlapping Bernoulli product distribution. We derive explicit formulas for the expectation, variance, and the generating function of higher moments, and formulate a conjecture asserting that the Shannon entropy is concave. The conjecture is proved in the two extremal cases and supported by symbolic computations for numerous overlap sequences.

cs.IT↗