相关论文: Improved upper bounds on the number of non-zero we…

For any simple-root constacyclic code $\mathcal{C}$ over a finite field $\mathbb{F}_q$, as far as we know, the group $\mathcal{G}$ generated by the multiplier, the constacyclic shift and the scalar multiplications is the largest subgroup of…

Let $\mathcal{C}$ be a quasi-cyclic code of index $l(l\geq2)$. Let $G$ be the subgroup of the automorphism group of $\mathcal{C}$ generated by $\rho^l$ and the scalar multiplications of $\mathcal{C}$, where $\rho$ denotes the standard…

For a simple-root $\lambda$-constacyclic code $\mathcal{C}$ over $\mathbb{F}_q$, let $\langle\rho\rangle$ and $\langle\rho,M\rangle$ be the subgroups of the automorphism group of $\mathcal{C}$ generated by the cyclic shift $\rho$, and by…

Upper and lower bounds on the largest number of weights in a cyclic code of given length, dimension and alphabet are given. An application to irreducible cyclic codes is considered. Sharper upper bounds are given for the special cyclic…

In this paper, the cyclic codes of length $n$, where $n$ is odd with certain restrictions, over a finite chain ring $R$, have been studied using the structure of group algebra approach. The primitive idempotents of $RG$ of a finite cyclic…

Let $X$ be a compact Riemann surface of genus $g\geq 2$, and let $G$ be a subgroup of $Aut(X)$. We show that if the Sylow $2$-subgroups of $G$ are cyclic, then $|G|\leq 30(g-1)$. If all Sylow subgroups of $G$ are cyclic, then, with two…

We focus on two aspects of cyclic orbit codes: invariants under equivalence and quasi-optimality. Regarding the first aspect, we establish a connection between the codewords of a cyclic orbit code and a certain linear set on the projective…

It is well known that the problem of determining the weight distributions of families of cyclic codes is, in general, notoriously difficult. An even harder problem is to find characterizations of families of cyclic codes in terms of their…

For a subshift over a finite alphabet, a measure of the complexity of the system is obtained by counting the number of nonempty cylinder sets of length $n$. When this complexity grows exponentially, the automorphism group has been shown to…

We study finite groups $G$ such that the maximum length of an orbit of the natural action of the automorphism group $\operatorname{Aut}(G)$ on $G$ is bounded from above by a constant. Our main results are the following: Firstly, a finite…

We determine a condition on the minimum Hamming weight of some special abelian group codes and, as a consequence of this result, we establish that any such code is, up to permutational equivalence, a subspace of the direct sum of $s$ copies…

Based on cyclic and consta-cyclic simplex codes, a new explicit construction of a family of two-weight codes is presented. These two-weight codes obtained are in the form of 2-generator quasi-cyclic, or quasi-twisted structure. Based on…

The problem of finding upper bounds for minimal vertex number of graphs with a given automorphism group is addressed in this article for the case of cyclic $2$-groups. We show that for any natural $n\ge 2$ there is an undirected graph…

An upper bound for the number of Hamiltonian cycles of symmetric diagraphs is established first in this paper, which is tighter than the famous Minc's bound and the Br$\acute{e}$gman's bound. A transformation on graphs is proposed, so that…

Cyclic codes with a few weights are very useful in the design of frequency hopping sequences and the development of secret sharing schemes. In this paper, we mainly use Gauss sums to represent the Hamming weights of a general construction…

An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that any two distinct vertices of $G$ have distinct closed neighbourhoods within $C$. These codes have been widely studied for over two decades. We give an improvement…

Given a finite non-cyclic group $G$, call $\sigma(G)$ the least number of proper subgroups of $G$ needed to cover $G$. In this paper we give lower and upper bounds for $\sigma(G)$ for $G$ a group with a unique minimal normal subgroup $N$…

Hopping cyclic codes (HCCs) are (non-linear) cyclic codes with the additional property that the $n$ cyclic shifts of every given codeword are all distinct, where $n$ is the code length. Constant weight binary hopping cyclic codes are also…

For a linear Hamming metric code of length n over a finite field, the number of distinct weights of its codewords is at most n. The codes achieving the equality in the above bound were called full weight spectrum codes. In this paper we…

Usually, it is difficult to determine the weight distribution of an irreducible cyclic code. In this paper, we discuss the case when an irreducible cyclic code has the maximal number of distinct nonzero weights and give a necessary and…