Related papers: Plotkin construction: rank and kernel
Polar codes have received increasing attention in the past decade, and have been selected for the next generation of wireless communication standard. Most research on polar codes has focused on codes constructed from a $2\times2$…
Few-weight codes over finite chain rings are associated with combinatorial objects such as strongly regular graphs (SRGs), strongly walk-regular graphs (SWRGs) and finite geometries, and are also widely used in data storage systems and…
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…
We consider an inverse problem for the Boltzmann equation with nonlinear collision operator in dimensions $n\geq 2$. We show that the kinetic collision kernel can be uniquely determined from the incoming-to-outgoing mappings on the boundary…
Codes in the sum-rank metric have received many attentions in recent years, since they have wide applications in the multishot network coding, the space-time coding and the distributed storage. In this paper, by constructing covering codes…
The hull of a linear code $C$ is the intersection of $C$ with its dual code. We present and analyze the number of linear $q$-ary codes of the same length and dimension but with different dimensions for their hulls. We prove that for given…
We study the rank weight hierarchy, thus in particular the rank metric, of cyclic codes over the finite field $\mathbb F_{q^m}$, $q$ a prime power, $m \geq 2$. We establish the rank weight hierarchy for $[n,n-1]$ cyclic codes and…
In this note, we reveal a relation between the weight distribution of a concatenated code ensemble based on the Plotkin construction and those of its component codes. The relation may find applications in the calculation of the ensemble…
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…
The structure of a linear block code is pivotal in defining fundamental properties, particularly weight distribution, and code design. In this study, we characterize the Type II structure of polar codewords with weights less than twice the…
In this text we develop the formalism of products and powers of linear codes under componentwise multiplication. As an expanded version of the author's talk at AGCT-14, focus is put mostly on basic properties and descriptive statements that…
Consider the problem of constructing a polar code of block length $N$ for the transmission over a given channel $W$. Typically this requires to compute the reliability of all the $N$ synthetic channels and then to include those that are…
Binary matrix codes with restricted row and column weights are a desirable method of coded modulation for power line communication. In this work, we construct such matrix codes that are obtained as products of affine codes - cosets of…
For each rank metric code $\mathcal{C}\subseteq \mathbb{K}^{m\times n}$, we associate a translation structure, the kernel of which is shown to be invariant with respect to the equivalence on rank metric codes. When $\mathcal{C}$ is…
A (q,k,t)-design matrix is an m x n matrix whose pattern of zeros/non-zeros satisfies the following design-like condition: each row has at most q non-zeros, each column has at least k non-zeros and the supports of every two columns…
Nowadays there are several classes of constrained codes intended for different applications. The following two large classes can be distinguished. The first class contains codes with local constraints; for example, the source data must be…
In this paper we investigate connections between linear sets and subspaces of linear maps. We give a geometric interpretation of the results of [18, Section 5] on linear sets on a projective line. We extend this to linear sets in arbitrary…
Sum-rank metric codes are a natural extension of both linear block codes and rank-metric codes. They have several applications in information theory, including multishot network coding and distributed storage systems. The aim of this…
We show that the sequence of dimensions of the linear spaces, generated by a given rank-metric code together with itself under several applications of a field automorphism, is an invariant for the whole equivalence class of the code. The…
The linkage construction and its generalization is one of the most powerful constructions for constant dimension code, accounting for approximately 50\% of all the listed parameters. We show how to improve the linkage construction of…