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Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Inspired by the recent work on binary cyclic codes published in Tang and Ding (2022) [26] and the works in Liu et al. (2023) [15] and Liu et al. (2023) [16] , the objectives of this paper are the construction and analysis of four infinite families of ternary cyclic codes with length n = 3 m − 1 for odd m and dimension k ∈ { n / 2 , (n + 2) / 2 } whose minimum distances have a square-root-like lower bound. Their duals have parameters [ n , k ⊥ , d ⊥ ] , where k ⊥ ∈ { n / 2 , (n − 2) / 2 } and d ⊥ also has a square-root-like lower bound. These families of codes and their duals contain distance-optimal cyclic codes. [ABSTRACT FROM AUTHOR]

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 p ≥ 5 be an odd prime and m be a positive integer. Let C (1 , e , s) denote the p -ary cyclic code with three nonzeros α , α e , and α s , where α is a generator of F p m ⁎ , s = p m − 1 2 , and 2 ≤ e ≤ p m − 2. In this paper, by analyzing the solutions of certain equations over finite fields, we present four classes of optimal p -ary cyclic codes C (1 , e , s) with parameters [ p m − 1 , p m − 2 m − 2 , 4 ]. Some known results on optimal quinary cyclic codes with parameters [ 5 m − 1 , 5 m − 2 m − 2 , 4 ] are special cases of our constructions. In addition, by analyzing the irreducible factors of certain polynomials over F 5 m , we present two classes of optimal quinary cyclic codes C (1 , e , s). [ABSTRACT FROM AUTHOR]

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 2013, Ding, et al. presented nine open problems about optimal ternary cyclic codes. The first two and the 6th problems were completely solved, and the third and last three problems were partially solved. In this paper, we focus on the 7th problem. By determining the root set of some special polynomials over finite fields, we give an incomplete answer and then present a counterexample. Furthermore, we construct two new classes of optimal ternary cyclic codes with respect to the Sphere Packing Bound by studying some special polynomials over finite fields. [ABSTRACT FROM AUTHOR]

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, we settle an open problem about a class of optimal ternary cyclic codes which was proposed by Ding and Helleseth [5]. Let C (1 , e) be a cyclic code of length 3 m − 1 over GF (3) with two nonzeros α and α e , where m and e are given integers, and α is a generator of GF (3 m) ⁎. It is shown that C (1 , e) is optimal with parameters [ 3 m − 1 , 3 m − 1 − 2 m , 4 ] if one of the following conditions is met. 1) m ≡ 0 (mod 4) , m ≥ 4 , and e = 3 m 2 + 5. 2) m ≡ 2 (mod 4) , m ≥ 6 , and e = 3 m + 2 2 + 5. [ABSTRACT FROM AUTHOR]

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Inspired by the recent work on binary cyclic codes published in Tang and Ding (2022) [14] , the objectives of this paper are the construction and analyses of five infinite families of binary cyclic codes with parameters [ n , k ] and (n − 6) / 3 ≤ k ≤ 2 (n + 6) / 3. Three of the five families of binary cyclic codes and their duals have a very good lower bound on their minimum distances and contain distance-optimal codes. The other two families of binary cyclic codes are composed of binary duadic codes with a square-root-like lower bound on their minimum distances. As a by-product, two infinite families of self-dual binary codes with a square-root-like lower bound on their minimum distances are obtained. [ABSTRACT FROM AUTHOR]

Reed-Solomon codes have gained a lot of interest due to its encoding simplicity, well structuredness and list-decoding capability [6] in the classical setting. This interest also translates to other metric setting, including the insertion and deletion (insdel for short) setting which is used to model synchronization errors caused by positional information loss in communication systems. Such interest is supported by the construction of a deletion correcting algorithm of insdel Reed-Solomon code in [22] which is based on the Guruswami-Sudan decoding algorithm [6]. Nevertheless, there have been few studies [3] on the insdel error-correcting capability of Reed-Solomon codes. In this paper, we discuss a criterion for a 2-dimensional insdel Reed-Solomon codes to have optimal asymptotic error-correcting capabilities, which are up to their respective lengths. Then we provide explicit constructions of 2-dimensional insdel Reed-Solomon codes that satisfy the established criteria. The family of such constructed codes can then be shown to extend the family of codes with asymptotic error-correcting capability reaching their respective lengths provided in [3, Theorem 2] which provide larger error-correcting capability compared to those defined in [25]. [ABSTRACT FROM AUTHOR]

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. The objective of this paper is to investigate the parameters of some q -ary cyclic codes of length n = q 2 m − 1 q + 1. The dimensions of these codes are settled and the lower bounds on their minimum distances are presented by employing the Hartmann-Tzeng bound or BCH bound. Moreover, when q = 2 , 3 , we give the largest coset leaders δ 1 modulo n and investigate the parameters of the BCH codes C (q , n , δ 1 , 1). [ABSTRACT FROM AUTHOR]