Related papers: Shortened Linear Codes over Finite Fields
Random linear codes (RLCs) are well known to have nice combinatorial properties and near-optimal parameters in many different settings. However, getting explicit constructions matching the parameters of RLCs is challenging, and RLCs are…
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…
In this work, we address the low-complexity construction of shortened and punctured polar codes from a unified view. While several independent puncturing and shortening designs were attempted in the literature, our goal is a unique,…
Currently known secondary construction techniques for linear codes mainly include puncturing, shortening, and extending. In this paper, we propose a novel method for the secondary construction of linear codes based on their weight…
The classical way of extending an $[n, k, d]$ linear code $\C$ is to add an overall parity-check coordinate to each codeword of the linear code $\C$. This extended code, denoted by $\overline{\C}(-\bone)$ and called the standardly extended…
The problem of identifying whether the family of cyclic codes is asymptotically good or not is a long-standing open problem in the field of coding theory. It is known in the literature that some families of cyclic codes such as BCH codes…
Polar codes were originally specified for codelengths that are powers of two. In many applications, it is desired to have a code that is not restricted to such lengths. Two common strategies of modifying the length of a code are shortening…
Linear codes are widely studied in coding theory as they have nice applications in distributed storage, combinatorics, lattices, cryptography and so on. Constructing linear codes with desirable properties is an interesting research topic.…
Random linear codes are a workhorse in coding theory, and are used to show the existence of codes with the best known or even near-optimal trade-offs in many noise models. However, they have little structure besides linearity, and are not…
Two general constructions of linear codes with functions over finite fields have been extensively studied in the literature. The first one is given by $\mathcal{C}(f)=\left\{ {\rm Tr}(af(x)+bx)_{x \in \mathbb{F}_{q^m}^*}: a,b \in…
A generic construction of linear codes over finite fields has recently received a lot of attention, and many one-weight, two-weight and three-weight codes with good error correcting capability have been produced with this generic approach.…
Linear error-correcting codes form the mathematical backbone of modern digital communication and storage systems, but identifying champion linear codes (linear codes achieving or exceeding the best known minimum Hamming distance) remains…
A new construction of codes from old ones is considered, it is an extension of the matrix-product construction. Several linear codes that improve the parameters of the known ones are presented.
Constacyclic codes over finite fields are a family of linear codes and contain cyclic codes as a subclass. Constacyclic codes are related to many areas of mathematics and outperform cyclic codes in several aspects. Hence, constacyclic codes…
Linear codes with a few weights have many nice applications including combinatorial design, distributed storage system, secret sharing schemes and so on. In this paper, we construct two families of linear codes with a few weights based on…
Linear codes have diverse applications in secret sharing schemes, secure two-party computation, association schemes, strongly regular graphs, authentication codes and communication. There are a large number of linear codes with few weights…
Sixteen new linear codes are presented: three of them improve the lower bounds on the minimum distance for a linear code and the rest are an explicit construction of unknown codes attaining the lower bounds on the minimum distance. They are…
Locally recoverable codes (LRCs) were proposed for the recovery of data in distributed and cloud storage systems about nine years ago. A lot of progress on the study of LRCs has been made by now. However, there is a lack of general theory…
This paper deals with the problem of increasing the minimum distance of a linear code by adding one or more columns to the generator matrix. Several methods to compute extensions of linear codes are presented. Many codes improving the…
This paper examines linear binary codes capable of correcting one or more errors. For the single-error-correcting case, it is shown that the Hamming bound is achieved by a constructive method, and an exact expression for the minimal…