Five Families of Three-Weight Ternary Cyclic Codes and Their Duals

As a subclass of linear codes, cyclic codes have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, five families of three-weight ternary cyclic codes whose duals have two zeros are presented. The weight distributions of the five families of cyclic codes are settled. The duals of two families of the cyclic codes are optimal

A Class of Three-Weight Cyclic Codes

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, a class of three-weight cyclic codes over \gf(p) whose duals have two zeros is presented, where $p$ is an odd prime. The weight distribution of this class of cyclic codes is settled. Some of the cyclic codes are optimal. The duals of a subclass of the cyclic codes are also studied and proved to be optimal.Comment: 11 Page

On the weight distributions of several classes of cyclic codes from APN monomials

Let $m\geq 3$ be an odd integer and $p$ be an odd prime. % with $p-1=2^rh$, where $h$ is an odd integer. In this paper, many classes of three-weight cyclic codes over $\mathbb{F}_{p}$ are presented via an examination of the condition for the cyclic codes $\mathcal{C}_{(1,d)}$ and $\mathcal{C}_{(1,e)}$, which have parity-check polynomials $m_1(x)m_d(x)$ and $m_1(x)m_e(x)$ respectively, to have the same weight distribution, where $m_i(x)$ is the minimal polynomial of $\pi^{-i}$ over $\mathbb{F}_{p}$ for a primitive element $\pi$ of $\mathbb{F}_{p^m}$. %For $p=3$, the duals of five classes of the proposed cyclic codes are optimal in the sense that they meet certain bounds on linear codes. Furthermore, for $p\equiv 3 \pmod{4}$ and positive integers $e$ such that there exist integers $k$ with $\gcd(m,k)=1$ and $\tau\in\{0,1,\cdots, m-1\}$ satisfying $(p^k+1)\cdot e\equiv 2 p^{\tau}\pmod{p^m-1}$, the value distributions of the two exponential sums T(a,b)=\sum\limits_{x\in \mathbb{F}_{p^m}}\omega^{\Tr(ax+bx^e)} and S(a,b,c)=\sum\limits_{x\in \mathbb{F}_{p^m}}\omega^{\Tr(ax+bx^e+cx^s)}, where $s=(p^m-1)/2$, are settled. As an application, the value distribution of $S(a,b,c)$ is utilized to investigate the weight distribution of the cyclic codes $\mathcal{C}_{(1,e,s)}$ with parity-check polynomial $m_1(x)m_e(x)m_s(x)$. In the case of $p=3$ and even $e$ satisfying the above condition, the duals of the cyclic codes $\mathcal{C}_{(1,e,s)}$ have the optimal minimum distance

The Weight Enumerator of Three Families of Cyclic Codes

Cyclic codes are a subclass of linear codes and have wide applications in consumer electronics, data storage systems, and communication systems due to their efficient encoding and decoding algorithms. Cyclic codes with many zeros and their dual codes have been a subject of study for many years. However, their weight distributions are known only for a very small number of cases. In general the calculation of the weight distribution of cyclic codes is heavily based on the evaluation of some exponential sums over finite fields. Very recently, Li, Hu, Feng and Ge studied a class of $p$-ary cyclic codes of length $p^{2m}-1$, where $p$ is a prime and $m$ is odd. They determined the weight distribution of this class of cyclic codes by establishing a connection between the involved exponential sums with the spectrum of Hermitian forms graphs. In this paper, this class of $p$-ary cyclic codes is generalized and the weight distribution of the generalized cyclic codes is settled for both even $m$ and odd $m$ alone with the idea of Li, Hu, Feng, and Ge. The weight distributions of two related families of cyclic codes are also determined.Comment: 13 Pages, 3 Table