Related papers: Function-Correcting Codes for Linear and Locally B…
This paper computationally obtains optimal bounded-weight, binary, error-correcting codes for a variety of distance bounds and dimensions. We compare the sizes of our codes to the sizes of optimal constant-weight, binary, error-correcting…
As an important coding scheme in modern distributed storage systems, locally repairable codes (LRCs) have attracted a lot of attentions from perspectives of both practical applications and theoretical research. As a major topic in the…
We consider the problem of designing [n; k] linear codes for distributed storage systems (DSS) that satisfy the (r, t)-Local Repair Property, where any t'(<=t) simultaneously failed nodes can be locally repaired, each with locality r. The…
Recent studies have delved into the construction of locally repairable codes (LRCs) with optimal minimum distance from function fields. In this paper, we present several novel constructions by extending the findings of optimally designed…
The local (central) limit theorem precisely describes the behavior of iterated convolution powers of a probability distribution on the $d$-dimensional integer lattice, $\mathbb{Z}^d$. Under certain mild assumptions on the distribution, the…
Linear codes generated by component functions of perfect nonlinear (PN) and almost perfect nonlinear (APN) functions and the first-order Reed-Muller codes have been an object of intensive study in coding theory. The objective of this paper…
The sum-rank metric naturally extends both the Hamming and rank metrics in coding theory over fields. It measures the error-correcting capability of codes in multishot matrix-multiplicative channels (e.g. linear network coding or the…
In this paper, we apply two-to-one functions over $\mathbb{F}_{2^n}$ in two generic constructions of binary linear codes. We consider two-to-one functions in two forms: (1) generalized quadratic functions; and (2) $\left(x^{2^t}+x\right)^e$…
In coding theory, handling errors that occur when symbols are inserted or deleted from a transmitted message is a long-standing challenge. Optimising redundancy for insertion and deletion channels remains a key open problem with significant…
In this paper, we present error-correcting codes which are the results of our research on the sub-exceeding functions. For a short and medium distance data transmission (wifi network, bluetooth, cable, ...), we see that these codes…
We derive a linear programming bound on the maximum cardinality of error-correcting codes in the sum-rank metric. Based on computational experiments on relatively small instances, we observe that the obtained bounds outperform all…
We prove the Local Limit Theorems for bounded additive functionals of uniformly elliptic inhomogeneous Markov arrays. As an application we obtain the precise asymptotics in the large deviation regime for bounded additive functionals of…
A geometrically local quantum code is an error correcting code situated within $\mathbb{R}^D$, where the checks only act on qubits within a fixed spatial distance. The main question is: What is the optimal dimension and distance for a…
Locally repairable codes have been investigated extensively in recent years due to practical applications in distributed and cloud storage systems. However, there are few asymptotical constructions of locally repairable codes in the…
Quantum information is fragile and must be protected by a quantum error-correcting code for large-scale practical applications. Recently, highly efficient quantum codes have been discovered which require a high degree of spatial…
We introduce tile codes, a simple yet powerful way of constructing quantum codes that are local on a planar 2D-lattice. Tile codes generalize the usual surface code by allowing for a bit more flexibility in terms of locality and stabilizer…
In this two-part study we develop a unified approach to the analysis of the global exactness of various penalty and augmented Lagrangian functions for finite-dimensional constrained optimization problems. This approach allows one to verify…
Linear codes play a central role in coding theory and have applications in several branches of mathematics. For error correction purposes the minimum Hamming distance should be as large as possible. Linear codes related to applications in…
When a node in a distributed storage system fails, it needs to be promptly repaired to maintain system integrity. While typical erasure codes can provide a significant storage advantage over replication, they suffer from poor repair…
We compute structure functions in the Hamiltonian formalism on a momentum lattice using a physically motivated regularisation that links the maximal parton number to the lattice size. We show for the $\phi ^4 _{3+1}$ theory that our method…