Related papers: Transitive nonpropelinear perfect codes

A lot of attention has been paid to the investigation of the algebraic properties of linear codes. In most cases, this investigation involves the determination of required code automorphisms, which are useful for decoders, such as the…

Let $R$ be a finite commutative chain ring with unique maximal ideal $\langle \gamma\rangle$, and let $n$ be a positive integer coprime with the characteristic of $R/\langle \gamma\rangle$. In this paper, the algebraic structure of cyclic…

Let $R=\mathbb{Z}_{4}[v]/\langle v^2+2v\rangle=\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ ($v^2=2v$) and $n$ be an odd positive integer. Then $R$ is a local non-principal ideal ring of $16$ elements and there is a $\mathbb{Z}_{4}$-linear Gray map from…

Non-overlapping codes are block codes that have arisen in diverse contexts of computer science and biology. Applications typically require finding non-overlapping codes with large cardinalities, but the maximum size of non-overlapping codes…

The projective general linear group $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ acts as a $3$-transitive permutation group on the set of points of the projective line. The first objective of this paper is to prove that all linear codes over…

We say that a loop is unbreakable when it does not have nontrivial subloops. While the cyclic groups of prime order are the only unbreakable finite groups, we show that nonassociative unbreakable loops exist for every order n >= 5. We…

Recently it was proved that if a linear code is invariant under the action of a doubly transitive permutation group, it achieves the capacity of erasure channel. Therefore, it is of sufficient interest to classify all codes, invariant under…

Code completion is an essential feature of IDEs, yet current autocompleters are restricted to either grammar-based or NLP-based single token completions. Both approaches have significant drawbacks: grammar-based autocompletion is restricted…

The minimum distance graph of a code has the codewords as vertices and edges exactly when the Hamming distance between two codewords equals the minimum distance of the code. A constructive proof for reconstructibility of an extended perfect…

Subspace codes have received an increasing interest recently due to their application in error-correction for random network coding. In particular, cyclic subspace codes are possible candidates for large codes with efficient encoding and…

The permutation groups of cyclic codes are widely applicable in determining the weight distribution of codes, decoding theory and various other areas. In this paper, by employing two distinct matrix representations, we can relate cyclic…

We prove that every countable acylindrically hyperbolic group admits a highly transitive action with finite kernel. This theorem uniformly generalizes many previously known results and allows us to answer a question of Garion and Glassner…

We study the asymptotic behavior of a family of algebraic geometry codes which are 4-quasi transitive linear codes. We prove that this family is asymptotically good over many prime fields using towers of algebraic function fields.

This paper investigates the application of the theoretical algebraic notion of a separable ring extension, in the realm of cyclic convolutional codes or, more generally, ideal codes. We work under very mild conditions, that cover all…

We examine the different ways of writing a cover of curves $\phi\colon C\to D$ over a field $K$ as a composition $\phi=\phi_n\circ\phi_{n-1}\circ\dots\circ\phi_1$, where each $\phi_i$ is a cover of curves over $K$ of degree at least $2$…

A subset $C$ of the vertex set $V$ of a graph $\Gamma$ is called a perfect code in $\Gamma$ if every vertex in $V\setminus C$ is adjacent to exactly one vertex in $C$. Given a group $G$ and a subgroup $H$ of $G$, a subgroup $A$ of $G$…

We call a linear code $C$ with length $n$ over a field $F$, a linear complementary equi-dual code, when there exists a linear code $D$ over $F$ such that $D$ is permutation equivalent to $C^\perp$ and $(C,D)$ is a linear complementary pair…

A multifold $1$-perfect code ($1$-perfect code for list decoding) in any graph is a set $C$ of vertices such that every vertex of the graph is at distance not more than $1$ from exactly $\mu$ elements of $C$. In $q$-ary Hamming graphs,…

A general class of the almost instantaneous fixed-to-variable-length (AIFV) codes is proposed, which contains every possible binary code we can make when allowing finite bits of decoding delay. The contribution of the paper lies in the…

We construct explicitly two infinite families of genuine nonadditive 1-error correcting quantum codes and prove that their coding subspaces are 50% larger than those of the optimal stabilizer codes of the same parameters via the linear…