Sensors measuring real-life physical processes are ubiquitous in today's interconnected world. These sensors inherently bear noise that often adversely affects the performance and reliability of the systems they support. Classic filtering approaches introduce strong assumption on the time or frequency characteristics of sensory measurements, while learning-based denoising approaches typically rely on using ground truth clean data to train a denoising model, which is often challenging or prohibitive to obtain for many real-world applications. We observe that in many scenarios, the relationships between different sensor measurements (e.g., location and acceleration) are analytically described by laws of physics (e.g., second-order differential equation). By incorporating such physics constraints, we can guide the denoising process to improve performance even in the absence of ground truth data. In light of this, we design a physics-informed denoising model that leverages the inherent algebraic relationships between different measurements governed by the underlying physics. By obviating the need for ground truth clean data, our method offers a practical denoising solution for real-world applications. We conducted experiments in various domains, including inertial navigation, CO2 monitoring, and HVAC control, and achieved state-of-the-art performance compared with existing denoising methods. Our method can denoise data in real time (4ms for a sequence of 1s) for low-cost noisy sensors and produces results that closely align with those from high-precision, high-cost alternatives, leading to an efficient, cost-effective approach for more accurate sensor-based systems.

For a linear block code C, its stopping redundancy is defined as the smallest number of check nodes in a Tanner graph for C, such that there exist no stopping sets of size smaller than the minimum distance of C. Schwartz and Vardy conjectured that the stopping redundancy of a maximum-distance separable (MDS) code should only depend on its length and minimum distance. We define the (n, t)-single-exclusion number, S(n, t) as the smallest number of i-subsets of an n-set, such that for each i-subset of the n-set, i =1,... ,t + 1, there exists a i-subset that contains all but one element of the i-subset. New upper bounds on the single-exclusion number are obtained via probabilistic methods, recurrent inequalities, as well as explicit constructions. The new bounds are used to better understand the stopping redundancy of MDS codes. In particular, it is shown that for [n, k = n - d + 1, d] MDS codes, as n rarr infin , the stopping redundancy is asymptotic to S(n, d - 2), if d = o(radic(n)), or if k = o(radic(n)), k rarr infin, thus giving partial confirmation of the Schwartz-Vardy conjecture in the asymptotic sense.

The ML redundancy of a code is defined as the smallest number of rows in its parity-check matrix such that a message-passing decoder working in the corresponding Tanner graph achieves maximum-likelihood (ML) performance on an erasure channel. General upper bounds on ML redundancy are obtained. In particular, it is shown that the ML redundancy of a q-ary code is at most the number of minimal codewords in its dual code, divided by q−1. Special upper bounds are derived for codes whose dual code contains a covering design. For example, the ML redundancy of a Simplex code of length n is shown to be no greater than (n2− 4n + 9)/6.

For a linear code C, the stopping redundancy of C is defined as the minimum number of check nodes in a Tanner graph T for C such that the size of the smallest stopping set in T is equal to the minimum distance of C. Han and Siegel recently proved an upper bound on the stopping redundancy of general linear codes, using probabilistic analysis. For most code parameters, this bound is the best currently known. In this correspondence, we present several improvements upon this bound.

We study memories protected with error control codes, in which the memory's contents are organized in lines which are read and written to in isolation from other lines. In these memories the available redundancy is structured so as to protect individual lines rather than the entire memory as a whole. Often designers wish to read and write only parts of the memory line, as in some instances this leads to various favorable system design tradeoffs, including better power consumption, increased data access concurrency, etc. (alternatively one may say that designers sometimes would prefer smaller line sizes). Nevertheless when designing systems with such subline accesses it is often found that in order to mantain a given level of reliability, the total amount of redundancy allocated in the memory needs to be increased beyond desirable levels. In this work, we initiate a study of the problem of structuring error control codes to allow subline accesses with good tradeoffs between reliability and redundancy. We motivate and explore a setting in which a "double-lookup" protocol is used in conjunction with certain types of two-level codes, whereby error detection is attained in a first level and error correction using the second level is performed whenever errors are detected in the first level. We obtain lower bounds on redundancy for a given level of reliability and offer a code construction that attains this bound for a certain important class of parameters. We also introduce an alternate construction which allows us to find longer codes under restrictions of the Galois field size used in the codes.

New bounds on single-exclusion numbers are obtained via probabilistic arguments, recurrent relations, as well as explicit constructions. The new bounds are used to better understand the stopping redundancy of MDS codes. In particular, it is shown that for any fixed k, the stopping redundancy of a linear [n, k] MDS code is between 1/k+1(kn) and (1 + o(1))1/k (kn).

For a linear block code with minimum distance d, its stopping redundancy is the minimum number of check nodes in a Tanner graph representation of the code, such that all nonempty stopping sets have size d or larger. We derive new upper bounds on stopping redundancy for all linear codes in general, and for maximum distance separable (MDS) codes specifically, and show how they improve upon previous results. For MDS codes, the new bounds are found by upper-bounding the stopping redundancy by a combinatorial quantity closely related to Turan numbers. (The Turan number, T(v,k,t), is the smallest number of t-subsets of a v-set, such that every k-subset of the v-set contains at least one of the t-subsets.) Asymptotically, we show that the stopping redundancy of MDS codes with length n and minimum distance d >1 is T(n,d-1,d-2)(1+O(n-1)) for fixed d, and is at most T (n,d-1,d-2)(3+O(n-1)) for fixed code dimension k=n-d+1. For d=3,4, we prove that the stopping redundancy of MDS codes is equal to T(n,d-1,d-2), for which exact formulas are known. For d=5, we show that the stopping redundancy of MDS codes is either T(n,4,3) or T(n,4,3)+1

The stopping redundancy of a linear code is defined as the minimum number of rows in its parity-check matrix such that the smallest stopping sets have size equal to the minimum distance of the code. We derive new upper bounds on the stopping redundancy of maximum distance separable (MDS) codes, and show how they improve upon previously known results. The new bounds are found by upper bounding the stopping redundancy by a combinatorial quantity closely related to Turan numbers. (The Turan number, T(v, k, t), is the smallest number of t-subsets of a v-set, such that every k-subset of the v-set contains at least one of the t-subsets.) Asymptotically, we show that the stopping redundancy of MDS codes with length n and minimum distance d > 1 is T(n, d -1, d - 2)(1 + O(n-1)) for fixed d, and is at most T(n, d - 1, d - 2)(3 + O(n-1)) for fixed code dimension k = n - d + 1. For d = 2,3,4, we prove that the stopping redundancy is equal to T(n, d - 1, d - 2). For d = 5, we show that the stopping redundancy is either T(n, 4, 3) or T(n, 4, 3) + 1

We derive upper and lower bounds on the weight distribution of Over-Extended Reed-Solomon (OERS) codes. Using these bounds, we obtain tight upper and lower bounds on the probability of undetected error for OERS codes on q-ary symmetric channels.

Upper and lower bounds on the weight distribution of overextended Reed-Solomon (OERS) codes are derived, from which tight upper and lower bounds on the probability of undetected error for OERS codes are obtained for q-ary symmetric channels