Related papers: Cyclic Codes and Sequences: the Generalized Kasami…

In this paper new binary sequence families $\mathcal{F}^k$ of period $2^n-1$ are constructed for even $n$ and any $k$ with ${\rm gcd}(k,n)=2$ if $n/2$ is odd or ${\rm gcd}(k,n)=1$ if $n/2$ is even. The distribution of their correlation…

Detailed information about the weight distribution of a convolutional code is given by the adjacency matrix of the state diagram associated with a controller canonical form of the code. We will show that this matrix is an invariant of the…

Let K be a p-adic field, R the valuation ring of K, and P the maximal ideal of R. Let $Y subseteq R^{2}$ be a non-singular closed curve, and Y_{m} its image in R/P^{m} times R/P^{m}, i.e. the reduction modulo P^{m} of Y. We denote by Psi an…

By exploiting the connection between scattered $\mathbb{F}_q$-subspaces of $\mathbb{F}_{q^m}^3$ and minimal non degenerate $3$-dimensional rank metric codes of $\mathbb{F}_{q^m}^{n}$, $n \geq m+2$, described in [2], we will exhibit a new…

The main objective of this paper is to extend the previously defined code family over the ring $\mathfrak{R}=\sum\limits_{s=0}^{4} v_{5}^{s} \mathcal{A}_{4}$ to $\mathfrak{R}^{s,m}=\sum\limits_{\varsigma=1}^{m}…

A cyclic codes of length $n$ over the rings $Z_{2^{m}}$ of integer of modulo $2^{m}$ is a linear code with property that if the codeword $(c_0,c_1,...,c_{n-1})\in \mathcal{C}$ then the cyclic shift $(c_1,c_2,...,c_0)\in \mathcal{C}$.…

Let omega(n) be the number of distinct prime factors dividing n and m > n natural numbers. We calculate a formula showing which prime numbers in which intervals divide a given binomial coefficient. From this formula we get an identity…

We study the distribution of cycle lengths in models of nonuniform random permutations with cycle weights. We identify several regimes. Depending on the weights, the length of typical cycles grows like the total number $n$ of elements, or a…

A class of optimal three-weight cyclic codes of dimension 3 over any finite field was presented by Vega [Finite Fields Appl., 42 (2016) 23-38]. Shortly thereafter, Heng and Yue [IEEE Trans. Inf. Theory, 62(8) (2016) 4501-4513] generalized…

In this paper, a family of six-weight cyclic codes over GF(p) whose duals have two zeros is presented, where p is an odd prime. And the weight distribution of these cyclic codes is determined.

Let $q$ be a power of $2$ and let $\mathbb{F}_q$ be the finite field with $q$ elements. For a positive integer $n$, the polynomial $X^n-1\in\mathbb{F}_q[X]$ is called $3$-sparse over $\mathbb{F}_q$ if every monic irreducible factor of…

This note is a stripped down version of a published paper on the Potts partition function, where we concentrate solely on the linear coding aspect of our approach. It is meant as a resource for people interested in coding theory but who do…

We investigate the weight distribution of random binary linear codes. For $0<\lambda<1$ and $n\to\infty$ pick uniformly at random $\lambda n$ vectors in $\mathbb{F}_2^n$ and let $C \le \mathbb{F}_2^n$ be the orthogonal complement of their…

In this note, we study skew cyclic and skew constacyclic codes over the ring $\mathcal{R}=F_{q}+uF_{q}+vF_{q}+uvF_{q}$ where $q=p^{m},$ $p$ is an odd prime, $u^{2}=u,~v^{2}=v,~uv=vu$. We show that Gray images of a skew cyclic and skew…

In this paper, we investigate the distribution of values of the complete exponential sum $S_{p,\chi}(\theta)=\sum_{n=1}^p \chi(n)e(n\theta)$, where $p$ is a large prime, $\chi$ is a Dirichlet character (mod $p$) of order $d\geq 2$, and…

There is a one-to-one correspondence between $\ell$-quasi-cyclic codes over a finite field $\mathbb F_q$ and linear codes over a ring $R = \mathbb F_q[Y]/(Y^m-1)$. Using this correspondence, we prove that every $\ell$-quasi-cyclic self-dual…

Cyclotomic polynomials are basic objects in Number Theory. Their properties depend on the number of distinct primes that intervene in the factorization of their order, and the binary case is thus the first nontrivial case. This paper sees…

We derive weighted sums, including binomial and double binomial sums, for the generalized Fibonacci sequence $\{G_m\}$ where for $m\ge 2$, $G_m=G_{m-1}+G_{m-2}$ with initial values $G_0$ and $G_1$.

We are interested in random uniform minimal factorizations of the $n$-cycle which are factorizations of $(1~2\dots n)$ into a product of $n-1$ transpositions. Our main result is an explicit formula for the joint probability that 1 and 2…

BCH codes are an important class of cyclic codes, and have wide applicantions in communication and storage systems. However, it is difficult to determine the parameters of BCH codes and only a few cases are known. In this paper, we mainly…