Related papers: Minimal linear codes from characteristic functions
Boolean functions have important applications in cryptography and coding theory. Two famous classes of binary codes derived from Boolean functions are the Reed-Muller codes and Kerdock codes. In the past two decades, a lot of progress on…
Subfield codes of linear codes over finite fields have recently received much attention. Some of these codes are optimal and have applications in secrete sharing, authentication codes and association schemes. In this paper, the $q$-ary…
Basic algebraic and combinatorial properties of finite vector spaces in which individual vectors are allowed to have multiplicities larger than $ 1 $ are derived. An application in coding theory is illustrated by showing that multispace…
We consider linear cyclic codes with the locality property, or locally recoverable codes (LRC codes). A family of LRC codes that generalizes the classical construction of Reed-Solomon codes was constructed in a recent paper by I. Tamo and…
Using linear functional-based duality of modules, we generalize the syndrome decoding algorithm of linear codes over finite fields to those over finite commutative rings. Moreover, If the ring is local the algorithm is simplified by…
In this work the construction of LRC codes given in [6] is completed, in the case of even characteristic. A general construction is presented, that enables us to obtain linear LRC codes of large length $n \approx q^4$, dimension and…
Recently, minimal linear codes have been extensively studied due to their applications in secret sharing schemes, secure two-party computations, and so on. Constructing minimal linear codes violating the Ashikhmin-Barg condition and then…
We consider the problem of coded distributed computing where a large linear computational job, such as a matrix multiplication, is divided into $k$ smaller tasks, encoded using an $(n,k)$ linear code, and performed over $n$ distributed…
We consider locally repairable codes over small fields and propose constructions of optimal cyclic and linear codes in terms of the dimension for a given distance and length. Four new constructions of optimal linear codes over small fields…
The class of uniformly computable real functions with respect to a small subrecursive class of operators computes the elementary functions of calculus, restricted to compact subsets of their domains. The class of conditionally computable…
Recently, simplicial complexes are used in constructions of several infinite families of minimal and optimal linear codes by Hyun {\em et al.} Building upon their research, in this paper more linear codes over the ring $\mathbb{Z}_4$ are…
The capability of discretization of matrix elements in the problem of quadratic functional minimization with linear member built on matrix in N-dimensional configuration space with discrete coordinates is researched. It is shown, that…
We present efficient and practical algorithms for a large, distributed system of processors to achieve reliable computations in a secure manner. Specifically, we address the problem of computing a general function of several private inputs…
This paper investigates the concept of self-dual convolutional code. We derive the basic properties of this interesting class of codes and we show how some of the techniques to construct self-dual linear block codes generalize to self-dual…
Cyclic codes are an interesting family of linear codes since they have efficient decoding algorithms and contain optimal codes as subfamilies. Constructing infinite families of cyclic codes with good parameters is important in both theory…
A minimal code is a linear code where the only instance that a codeword has its support contained in the support of another codeword is when the codewords are scalar multiples of each other. Ashikhmin and Barg gave a sufficient condition…
In this paper, we will give the generic construction of a binary linear code of dimension $n+3$ and derive the necessary and sufficient conditions for the constructed code to be minimal. Using generic construction, a new family of minimal…
The area related to M. Liv\v{s}ic's characteristic matrix functions is too vast to be discussed in one paper and we selected for this article the problems which are close to our scientific interests. We discuss M.Liv\v{s}ic's results…
Within the framework of mappings between affine spaces, the notion of $n$-th polarization of a function will lead to an intrinsic characterization of polynomial functions. We prove that the characteristic features of derivations, such as…
We show how the theory of affine geometries over the ring ${\mathbb Z}/\langle q - 1\rangle$ can be used to understand the properties of toric and generalized toric codes over ${\mathbb F}_q$. The minimum distance of these codes is strongly…