Several new classes of optimal ternary cyclic codes with two or three zeros

Cyclic codes are a subclass of linear codes and have wide applications in data storage systems, communication systems and consumer electronics due to their efficient encoding and decoding algorithms. Let α be a generator of F3ᵐ^*, where m is a positive integer. Denote by C(i₁,i₂,⋯, iₜ) the cyclic code with generator polynomial m_αi₁(x)m_αi₂(x)⋯ m_αiₜ(x), where m_αⁱ(x) is the minimal polynomial of α ⁱ over F₃. In this paper, by analyzing the solutions of certain equations over finite fields, we present four classes of optimal ternary cyclic codes C(0,1,e) and C(1,e,s) with parameters [3ᵐ-1,3ᵐ-3m/2-2,4], where s=3ᵐ-1/2. In addition, by determining the solutions of certain equations and analyzing the irreducible factors of certain polynomials over F3ᵐ, we present four classes of optimal ternary cyclic codes C(2,e) and C(1,e) with parameters [3ᵐ-1,3ᵐ-2m-1,4]. We show that our new optimal cyclic codes are inequivalent to the known ones.