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

Cyclic codes have efficient encoding and decoding algorithms. The decoding error probability and the undetected error probability are usually bounded by or given from the weight distributions of the codes. Most researches are about the…

We derive a recursive formula determing the weight distribution of the [n=(q^m-1)/(q-1), n-m, 3] Hamming code H(m,q), when (m, q-1)=1. Here q is a prime power. The proof is based on Moisio's idea of using Pless power moment identity…

Let $p$ be a prime, and let $N$ be a positive integer such that $p$ is a primitive root modulo $N$. Define $q = p^e$, where $e = \phi(N)$, and let $\mathbb{F}_q$ be the finite field of order $q$ with $\mathbb{F}_p$ as its prime subfield.…

In this paper, we make some progress towards a well-known conjecture on the minimum weights of binary cyclic codes with two primitive nonzeros. We also determine the Walsh spectrum of $\Tr(x^d)$ over $\F_{2^{m}}$ in the case where $m=2t$,…

Recently, the weight distributions of the duals of the cyclic codes with two zeros have been obtained for several cases. In this paper we use the method developed before to solve one more special case. We make extensive use of standard…

Cyclic codes are an important class of linear codes, whose weight distribution have been extensively studied. So far, most of previous results obtained were for cyclic codes with no more than three zeros. Recently, \cite{Y-X-D12}…

In a previous paper, we derived a recursive formula determining the weight distributions of the [n=(q^m-1)/(q-1)] Hamming code H(m,q), when (m,q-1)=1. Here q is a prime power. We note here that the formula actually holds for any positive…

The dual of the Kasami code of length $q^2-1$, with $q$ a power of $2$, is constructed by concatenating a cyclic MDS code of length $q+1$ over $F_q$ with a Simplex code of length $q-1$. This yields a new derivation of the weight…

Let $\mathbb{F}_q$ be a finite field of order $q$ and integer $n\ge 1$. Let $\ell$ be a prime such that $\ell^k|(q-1)$ for some integer $k\ge 1$ and $\mu$ be an element of order $\ell^k$ in $\mathbb{F}_q$. In this paper, we determine the…

Let $\ell^m$ be a power with $\ell$ a prime greater than $3$ and $m$ a positive integer such that $3$ is a primitive root modulo $2\ell^m$. Let $\mathbb{F}_3$ be the finite field of order $3$, and let $\mathbb{F}$ be the…

Let $f$ be a real polynomial with irrational leading co-efficient. In this article, we derive distribution of $f(n)$ modulo one for all $n$ with at least three divisors and also we study distribution of $f(n)$ for all square-free $n$ with…

Determining the weight distribution of a linear code is a classical and fundamental topic in coding theory that has been extensively investigated. Repeated-root cyclic codes, which form a significant subclass of error-correcting codes, have…

In this expository paper we show how one can, in a uniform way, calculate the weight distributions of some well-known binary cyclic codes. The codes are related to certain families of curves, and the weight distributions are related to the…

For an odd prime $p$ and two positive integers $n\geq 3$ and $k$ with $\frac{n}{{\rm gcd}(n,k)}$ being odd, the paper determines the weight distribution of a $p$-ary cyclic code $\mathcal{C}$ over $\mathbb{F}_{p}$ with nonzeros…

Let $q$ be an odd prime power and let $X(m,q)$ be the set of symmetric bilinear forms on an $m$-dimensional vector space over $\mathbb{F}_q$. The partition of $X(m,q)$ induced by the action of the general linear group gives rise to a…

In this article, we introduce and study the concept of the exponent of a cyclic code over a finite field $\mathbb{F}_q.$ We give a relation between the exponent of a cyclic code and its dual code. Finally, we introduce and determine the…

For sufficiently large integers $K$, $x$, $y$, and $q$ satisfying $K \le y < x$, where $f(u) = \alpha u^n + \alpha_{n-1}u^{n-1} + \ldots + \alpha_1 u$ is a polynomial of degree $n$ with real coefficients, $n$ is a fixed positive integer,…

Cyclic codes have attracted a lot of research interest for decades. In this paper, for an odd prime $p$, we propose a general strategy to compute the complete weight enumerator of cyclic codes via the value distribution of the corresponding…

Combinatorial $t$-designs have been an interesting topic in combinatorics for decades. It is a basic fact that the codewords of a fixed weight in a code may hold a $t$-design. Till now only a small amount of work on constructing $t$-designs…

In this paper, several families of irreducible constacyclic codes over finite fields and their duals are studied. The weight distributions of these irreducible constacyclic codes and the parameters of their duals are settled. Several…