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

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

Proof of a Conjectured Three-Valued Family of Weil Sums of Binomials

We consider Weil sums of binomials of the form $W_{F,d}(a)=\sum_{x \in F} \psi(x^d-a x)$, where $F$ is a finite field, $\psi\colon F\to {\mathbb C}$ is the canonical additive character, $\gcd(d,|F^\times|)=1$, and $a \in F^\times$. If we fix $F$ and $d$ and examine the values of $W_{F,d}(a)$ as $a$ runs through $F^\times$, we always obtain at least three distinct values unless $d$ is degenerate (a power of the characteristic of $F$ modulo $|F^\times|$). Choices of $F$ and $d$ for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if $F$ is a field of order $3^n$ with $n$ odd, and $d=3^r+2$ with $4 r \equiv 1 \pmod{n}$, then $W_{F,d}(a)$ assumes only the three values $0$ and $\pm 3^{(n+1)/2}$. This proves the 2001 conjecture of Dobbertin, Helleseth, Kumar, and Martinsen. The proof employs diverse methods involving trilinear forms, counting points on curves via multiplicative character sums, divisibility properties of Gauss sums, and graph theory.Comment: 19 page

A Family of Five-Weight Cyclic Codes and Their Weight Enumerators

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 family of $p$-ary cyclic codes whose duals have three zeros are proposed. The weight distribution of this family of cyclic codes is determined. It turns out that the proposed cyclic codes have five nonzero weights.Comment: 14 Page